Calculus Examples

$f (x) = {(x + 1)}^{3} {(x + 2)}^{2}$

Step 1

Find the second derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Rewrite ${(x + 2)}^{2}$ as $(x + 2) (x + 2)$ .

$\frac{d}{d x} [{(x + 1)}^{3} ((x + 2) (x + 2))]$

Step 1.1.2

Expand $(x + 2) (x + 2)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

$\frac{d}{d x} [{(x + 1)}^{3} (x (x + 2) + 2 (x + 2))]$

Step 1.1.2.2

Apply the distributive property.

$\frac{d}{d x} [{(x + 1)}^{3} (x \cdot x + x \cdot 2 + 2 (x + 2))]$

Step 1.1.2.3

Apply the distributive property.

$\frac{d}{d x} [{(x + 1)}^{3} (x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2)]$

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Multiply $x$ by $x$ .

$\frac{d}{d x} [{(x + 1)}^{3} (x^{2} + x \cdot 2 + 2 x + 2 \cdot 2)]$

Step 1.1.3.1.2

Move $2$ to the left of $x$ .

$\frac{d}{d x} [{(x + 1)}^{3} (x^{2} + 2 \cdot x + 2 x + 2 \cdot 2)]$

Step 1.1.3.1.3

Multiply $2$ by $2$ .

$\frac{d}{d x} [{(x + 1)}^{3} (x^{2} + 2 x + 2 x + 4)]$

Step 1.1.3.2

Add $2 x$ and $2 x$ .

$\frac{d}{d x} [{(x + 1)}^{3} (x^{2} + 4 x + 4)]$

Step 1.1.4

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = {(x + 1)}^{3}$ and $g (x) = x^{2} + 4 x + 4$ .

${(x + 1)}^{3} \frac{d}{d x} [x^{2} + 4 x + 4] + (x^{2} + 4 x + 4) \frac{d}{d x} [{(x + 1)}^{3}]$

Step 1.1.5

Differentiate.

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Step 1.1.5.1

By the Sum Rule, the derivative of $x^{2} + 4 x + 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [4]$ .

${(x + 1)}^{3} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [4]) + (x^{2} + 4 x + 4) \frac{d}{d x} [{(x + 1)}^{3}]$

Step 1.1.5.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

${(x + 1)}^{3} (2 x + \frac{d}{d x} [4 x] + \frac{d}{d x} [4]) + (x^{2} + 4 x + 4) \frac{d}{d x} [{(x + 1)}^{3}]$

Step 1.1.5.3

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

${(x + 1)}^{3} (2 x + 4 \frac{d}{d x} [x] + \frac{d}{d x} [4]) + (x^{2} + 4 x + 4) \frac{d}{d x} [{(x + 1)}^{3}]$

Step 1.1.5.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

${(x + 1)}^{3} (2 x + 4 \cdot 1 + \frac{d}{d x} [4]) + (x^{2} + 4 x + 4) \frac{d}{d x} [{(x + 1)}^{3}]$

Step 1.1.5.5

Multiply $4$ by $1$ .

${(x + 1)}^{3} (2 x + 4 + \frac{d}{d x} [4]) + (x^{2} + 4 x + 4) \frac{d}{d x} [{(x + 1)}^{3}]$

Step 1.1.5.6

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

${(x + 1)}^{3} (2 x + 4 + 0) + (x^{2} + 4 x + 4) \frac{d}{d x} [{(x + 1)}^{3}]$

Step 1.1.5.7

Add $2 x + 4$ and $0$ .

${(x + 1)}^{3} (2 x + 4) + (x^{2} + 4 x + 4) \frac{d}{d x} [{(x + 1)}^{3}]$

Step 1.1.6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = x + 1$ .

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Step 1.1.6.1

To apply the Chain Rule, set $u$ as $x + 1$ .

${(x + 1)}^{3} (2 x + 4) + (x^{2} + 4 x + 4) (\frac{d}{d u} [u^{3}] \frac{d}{d x} [x + 1])$

Step 1.1.6.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

${(x + 1)}^{3} (2 x + 4) + (x^{2} + 4 x + 4) (3 u^{2} \frac{d}{d x} [x + 1])$

Step 1.1.6.3

Replace all occurrences of $u$ with $x + 1$ .

${(x + 1)}^{3} (2 x + 4) + (x^{2} + 4 x + 4) (3 {(x + 1)}^{2} \frac{d}{d x} [x + 1])$

Step 1.1.7

Differentiate.

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Step 1.1.7.1

Move $3$ to the left of $x^{2} + 4 x + 4$ .

${(x + 1)}^{3} (2 x + 4) + 3 \cdot (x^{2} + 4 x + 4) {(x + 1)}^{2} \frac{d}{d x} [x + 1]$

Step 1.1.7.2

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

${(x + 1)}^{3} (2 x + 4) + 3 (x^{2} + 4 x + 4) {(x + 1)}^{2} (\frac{d}{d x} [x] + \frac{d}{d x} [1])$

Step 1.1.7.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

${(x + 1)}^{3} (2 x + 4) + 3 (x^{2} + 4 x + 4) {(x + 1)}^{2} (1 + \frac{d}{d x} [1])$

Step 1.1.7.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

${(x + 1)}^{3} (2 x + 4) + 3 (x^{2} + 4 x + 4) {(x + 1)}^{2} (1 + 0)$

Step 1.1.7.5

Simplify the expression.

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Step 1.1.7.5.1

Add $1$ and $0$ .

${(x + 1)}^{3} (2 x + 4) + 3 (x^{2} + 4 x + 4) {(x + 1)}^{2} \cdot 1$

Step 1.1.7.5.2

Multiply $3$ by $1$ .

${(x + 1)}^{3} (2 x + 4) + 3 (x^{2} + 4 x + 4) {(x + 1)}^{2}$

Step 1.1.8

Simplify.

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Step 1.1.8.1

Apply the distributive property.

${(x + 1)}^{3} (2 x + 4) + (3 x^{2} + 3 (4 x) + 3 \cdot 4) {(x + 1)}^{2}$

Step 1.1.8.2

Multiply $4$ by $3$ .

${(x + 1)}^{3} (2 x + 4) + (3 x^{2} + 12 x + 3 \cdot 4) {(x + 1)}^{2}$

Step 1.1.8.3

Multiply $3$ by $4$ .

${(x + 1)}^{3} (2 x + 4) + (3 x^{2} + 12 x + 12) {(x + 1)}^{2}$

Step 1.1.8.4

Factor ${(x + 1)}^{2}$ out of ${(x + 1)}^{3} (2 x + 4) + (3 x^{2} + 12 x + 12) {(x + 1)}^{2}$ .

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Step 1.1.8.4.1

Factor ${(x + 1)}^{2}$ out of ${(x + 1)}^{3} (2 x + 4)$ .

${(x + 1)}^{2} ((x + 1) (2 x + 4)) + (3 x^{2} + 12 x + 12) {(x + 1)}^{2}$

Step 1.1.8.4.2

Factor ${(x + 1)}^{2}$ out of $(3 x^{2} + 12 x + 12) {(x + 1)}^{2}$ .

${(x + 1)}^{2} ((x + 1) (2 x + 4)) + {(x + 1)}^{2} (3 x^{2} + 12 x + 12)$

Step 1.1.8.4.3

Factor ${(x + 1)}^{2}$ out of ${(x + 1)}^{2} ((x + 1) (2 x + 4)) + {(x + 1)}^{2} (3 x^{2} + 12 x + 12)$ .

${(x + 1)}^{2} ((x + 1) (2 x + 4) + 3 x^{2} + 12 x + 12)$

Step 1.1.8.5

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

$(x + 1) (x + 1) ((x + 1) (2 x + 4) + 3 x^{2} + 12 x + 12)$

Step 1.1.8.6

Expand $(x + 1) (x + 1)$ using the FOIL Method.

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Step 1.1.8.6.1

Apply the distributive property.

$(x (x + 1) + 1 (x + 1)) ((x + 1) (2 x + 4) + 3 x^{2} + 12 x + 12)$

Step 1.1.8.6.2

Apply the distributive property.

$(x \cdot x + x \cdot 1 + 1 (x + 1)) ((x + 1) (2 x + 4) + 3 x^{2} + 12 x + 12)$

Step 1.1.8.6.3

Apply the distributive property.

$(x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1) ((x + 1) (2 x + 4) + 3 x^{2} + 12 x + 12)$

Step 1.1.8.7

Simplify and combine like terms.

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Step 1.1.8.7.1

Simplify each term.

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Step 1.1.8.7.1.1

Multiply $x$ by $x$ .

$(x^{2} + x \cdot 1 + 1 x + 1 \cdot 1) ((x + 1) (2 x + 4) + 3 x^{2} + 12 x + 12)$

Step 1.1.8.7.1.2

Multiply $x$ by $1$ .

$(x^{2} + x + 1 x + 1 \cdot 1) ((x + 1) (2 x + 4) + 3 x^{2} + 12 x + 12)$

Step 1.1.8.7.1.3

Multiply $x$ by $1$ .

$(x^{2} + x + x + 1 \cdot 1) ((x + 1) (2 x + 4) + 3 x^{2} + 12 x + 12)$

Step 1.1.8.7.1.4

Multiply $1$ by $1$ .

$(x^{2} + x + x + 1) ((x + 1) (2 x + 4) + 3 x^{2} + 12 x + 12)$

Step 1.1.8.7.2

Add $x$ and $x$ .

$(x^{2} + 2 x + 1) ((x + 1) (2 x + 4) + 3 x^{2} + 12 x + 12)$

Step 1.1.8.8

Simplify each term.

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Step 1.1.8.8.1

Expand $(x + 1) (2 x + 4)$ using the FOIL Method.

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Step 1.1.8.8.1.1

Apply the distributive property.

$(x^{2} + 2 x + 1) (x (2 x + 4) + 1 (2 x + 4) + 3 x^{2} + 12 x + 12)$

Step 1.1.8.8.1.2

Apply the distributive property.

$(x^{2} + 2 x + 1) (x (2 x) + x \cdot 4 + 1 (2 x + 4) + 3 x^{2} + 12 x + 12)$

Step 1.1.8.8.1.3

Apply the distributive property.

$(x^{2} + 2 x + 1) (x (2 x) + x \cdot 4 + 1 (2 x) + 1 \cdot 4 + 3 x^{2} + 12 x + 12)$

Step 1.1.8.8.2

Simplify and combine like terms.

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Step 1.1.8.8.2.1

Simplify each term.

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Step 1.1.8.8.2.1.1

Rewrite using the commutative property of multiplication.

$(x^{2} + 2 x + 1) (2 x \cdot x + x \cdot 4 + 1 (2 x) + 1 \cdot 4 + 3 x^{2} + 12 x + 12)$

Step 1.1.8.8.2.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.8.8.2.1.2.1

Move $x$ .

$(x^{2} + 2 x + 1) (2 (x \cdot x) + x \cdot 4 + 1 (2 x) + 1 \cdot 4 + 3 x^{2} + 12 x + 12)$

Step 1.1.8.8.2.1.2.2

Multiply $x$ by $x$ .

$(x^{2} + 2 x + 1) (2 x^{2} + x \cdot 4 + 1 (2 x) + 1 \cdot 4 + 3 x^{2} + 12 x + 12)$

Step 1.1.8.8.2.1.3

Move $4$ to the left of $x$ .

$(x^{2} + 2 x + 1) (2 x^{2} + 4 \cdot x + 1 (2 x) + 1 \cdot 4 + 3 x^{2} + 12 x + 12)$

Step 1.1.8.8.2.1.4

Multiply $2 x$ by $1$ .

$(x^{2} + 2 x + 1) (2 x^{2} + 4 x + 2 x + 1 \cdot 4 + 3 x^{2} + 12 x + 12)$

Step 1.1.8.8.2.1.5

Multiply $4$ by $1$ .

$(x^{2} + 2 x + 1) (2 x^{2} + 4 x + 2 x + 4 + 3 x^{2} + 12 x + 12)$

Step 1.1.8.8.2.2

Add $4 x$ and $2 x$ .

$(x^{2} + 2 x + 1) (2 x^{2} + 6 x + 4 + 3 x^{2} + 12 x + 12)$

Step 1.1.8.9

Add $2 x^{2}$ and $3 x^{2}$ .

$(x^{2} + 2 x + 1) (5 x^{2} + 6 x + 4 + 12 x + 12)$

Step 1.1.8.10

Add $6 x$ and $12 x$ .

$(x^{2} + 2 x + 1) (5 x^{2} + 18 x + 4 + 12)$

Step 1.1.8.11

Add $4$ and $12$ .

$(x^{2} + 2 x + 1) (5 x^{2} + 18 x + 16)$

Step 1.1.8.12

Expand $(x^{2} + 2 x + 1) (5 x^{2} + 18 x + 16)$ by multiplying each term in the first expression by each term in the second expression.

$x^{2} (5 x^{2}) + x^{2} (18 x) + x^{2} \cdot 16 + 2 x (5 x^{2}) + 2 x (18 x) + 2 x \cdot 16 + 1 (5 x^{2}) + 1 (18 x) + 1 \cdot 16$

Step 1.1.8.13

Simplify each term.

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Step 1.1.8.13.1

Rewrite using the commutative property of multiplication.

$5 x^{2} x^{2} + x^{2} (18 x) + x^{2} \cdot 16 + 2 x (5 x^{2}) + 2 x (18 x) + 2 x \cdot 16 + 1 (5 x^{2}) + 1 (18 x) + 1 \cdot 16$

Step 1.1.8.13.2

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 1.1.8.13.2.1

Move $x^{2}$ .

$5 (x^{2} x^{2}) + x^{2} (18 x) + x^{2} \cdot 16 + 2 x (5 x^{2}) + 2 x (18 x) + 2 x \cdot 16 + 1 (5 x^{2}) + 1 (18 x) + 1 \cdot 16$

Step 1.1.8.13.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$5 x^{2 + 2} + x^{2} (18 x) + x^{2} \cdot 16 + 2 x (5 x^{2}) + 2 x (18 x) + 2 x \cdot 16 + 1 (5 x^{2}) + 1 (18 x) + 1 \cdot 16$

Step 1.1.8.13.2.3

Add $2$ and $2$ .

$5 x^{4} + x^{2} (18 x) + x^{2} \cdot 16 + 2 x (5 x^{2}) + 2 x (18 x) + 2 x \cdot 16 + 1 (5 x^{2}) + 1 (18 x) + 1 \cdot 16$

Step 1.1.8.13.3

Rewrite using the commutative property of multiplication.

$5 x^{4} + 18 x^{2} x + x^{2} \cdot 16 + 2 x (5 x^{2}) + 2 x (18 x) + 2 x \cdot 16 + 1 (5 x^{2}) + 1 (18 x) + 1 \cdot 16$

Step 1.1.8.13.4

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.1.8.13.4.1

Move $x$ .

$5 x^{4} + 18 (x \cdot x^{2}) + x^{2} \cdot 16 + 2 x (5 x^{2}) + 2 x (18 x) + 2 x \cdot 16 + 1 (5 x^{2}) + 1 (18 x) + 1 \cdot 16$

Step 1.1.8.13.4.2

Multiply $x$ by $x^{2}$ .

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Step 1.1.8.13.4.2.1

Raise $x$ to the power of $1$ .

$5 x^{4} + 18 (x^{1} x^{2}) + x^{2} \cdot 16 + 2 x (5 x^{2}) + 2 x (18 x) + 2 x \cdot 16 + 1 (5 x^{2}) + 1 (18 x) + 1 \cdot 16$

Step 1.1.8.13.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$5 x^{4} + 18 x^{1 + 2} + x^{2} \cdot 16 + 2 x (5 x^{2}) + 2 x (18 x) + 2 x \cdot 16 + 1 (5 x^{2}) + 1 (18 x) + 1 \cdot 16$

Step 1.1.8.13.4.3

Add $1$ and $2$ .

$5 x^{4} + 18 x^{3} + x^{2} \cdot 16 + 2 x (5 x^{2}) + 2 x (18 x) + 2 x \cdot 16 + 1 (5 x^{2}) + 1 (18 x) + 1 \cdot 16$

Step 1.1.8.13.5

Move $16$ to the left of $x^{2}$ .

$5 x^{4} + 18 x^{3} + 16 \cdot x^{2} + 2 x (5 x^{2}) + 2 x (18 x) + 2 x \cdot 16 + 1 (5 x^{2}) + 1 (18 x) + 1 \cdot 16$

Step 1.1.8.13.6

Rewrite using the commutative property of multiplication.

$5 x^{4} + 18 x^{3} + 16 x^{2} + 2 \cdot 5 x \cdot x^{2} + 2 x (18 x) + 2 x \cdot 16 + 1 (5 x^{2}) + 1 (18 x) + 1 \cdot 16$

Step 1.1.8.13.7

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 1.1.8.13.7.1

Move $x^{2}$ .

$5 x^{4} + 18 x^{3} + 16 x^{2} + 2 \cdot 5 (x^{2} x) + 2 x (18 x) + 2 x \cdot 16 + 1 (5 x^{2}) + 1 (18 x) + 1 \cdot 16$

Step 1.1.8.13.7.2

Multiply $x^{2}$ by $x$ .

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Step 1.1.8.13.7.2.1

Raise $x$ to the power of $1$ .

$5 x^{4} + 18 x^{3} + 16 x^{2} + 2 \cdot 5 (x^{2} x^{1}) + 2 x (18 x) + 2 x \cdot 16 + 1 (5 x^{2}) + 1 (18 x) + 1 \cdot 16$

Step 1.1.8.13.7.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$5 x^{4} + 18 x^{3} + 16 x^{2} + 2 \cdot 5 x^{2 + 1} + 2 x (18 x) + 2 x \cdot 16 + 1 (5 x^{2}) + 1 (18 x) + 1 \cdot 16$

Step 1.1.8.13.7.3

Add $2$ and $1$ .

$5 x^{4} + 18 x^{3} + 16 x^{2} + 2 \cdot 5 x^{3} + 2 x (18 x) + 2 x \cdot 16 + 1 (5 x^{2}) + 1 (18 x) + 1 \cdot 16$

Step 1.1.8.13.8

Multiply $2$ by $5$ .

$5 x^{4} + 18 x^{3} + 16 x^{2} + 10 x^{3} + 2 x (18 x) + 2 x \cdot 16 + 1 (5 x^{2}) + 1 (18 x) + 1 \cdot 16$

Step 1.1.8.13.9

Rewrite using the commutative property of multiplication.

$5 x^{4} + 18 x^{3} + 16 x^{2} + 10 x^{3} + 2 \cdot 18 x \cdot x + 2 x \cdot 16 + 1 (5 x^{2}) + 1 (18 x) + 1 \cdot 16$

Step 1.1.8.13.10

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.8.13.10.1

Move $x$ .

$5 x^{4} + 18 x^{3} + 16 x^{2} + 10 x^{3} + 2 \cdot 18 (x \cdot x) + 2 x \cdot 16 + 1 (5 x^{2}) + 1 (18 x) + 1 \cdot 16$

Step 1.1.8.13.10.2

Multiply $x$ by $x$ .

$5 x^{4} + 18 x^{3} + 16 x^{2} + 10 x^{3} + 2 \cdot 18 x^{2} + 2 x \cdot 16 + 1 (5 x^{2}) + 1 (18 x) + 1 \cdot 16$

Step 1.1.8.13.11

Multiply $2$ by $18$ .

$5 x^{4} + 18 x^{3} + 16 x^{2} + 10 x^{3} + 36 x^{2} + 2 x \cdot 16 + 1 (5 x^{2}) + 1 (18 x) + 1 \cdot 16$

Step 1.1.8.13.12

Multiply $16$ by $2$ .

$5 x^{4} + 18 x^{3} + 16 x^{2} + 10 x^{3} + 36 x^{2} + 32 x + 1 (5 x^{2}) + 1 (18 x) + 1 \cdot 16$

Step 1.1.8.13.13

Multiply $5 x^{2}$ by $1$ .

$5 x^{4} + 18 x^{3} + 16 x^{2} + 10 x^{3} + 36 x^{2} + 32 x + 5 x^{2} + 1 (18 x) + 1 \cdot 16$

Step 1.1.8.13.14

Multiply $18 x$ by $1$ .

$5 x^{4} + 18 x^{3} + 16 x^{2} + 10 x^{3} + 36 x^{2} + 32 x + 5 x^{2} + 18 x + 1 \cdot 16$

Step 1.1.8.13.15

Multiply $16$ by $1$ .

$5 x^{4} + 18 x^{3} + 16 x^{2} + 10 x^{3} + 36 x^{2} + 32 x + 5 x^{2} + 18 x + 16$

Step 1.1.8.14

Add $18 x^{3}$ and $10 x^{3}$ .

$5 x^{4} + 28 x^{3} + 16 x^{2} + 36 x^{2} + 32 x + 5 x^{2} + 18 x + 16$

Step 1.1.8.15

Add $16 x^{2}$ and $36 x^{2}$ .

$5 x^{4} + 28 x^{3} + 52 x^{2} + 32 x + 5 x^{2} + 18 x + 16$

Step 1.1.8.16

Add $52 x^{2}$ and $5 x^{2}$ .

$5 x^{4} + 28 x^{3} + 57 x^{2} + 32 x + 18 x + 16$

Step 1.1.8.17

Add $32 x$ and $18 x$ .

$f' (x) = 5 x^{4} + 28 x^{3} + 57 x^{2} + 50 x + 16$

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of $5 x^{4} + 28 x^{3} + 57 x^{2} + 50 x + 16$ with respect to $x$ is $\frac{d}{d x} [5 x^{4}] + \frac{d}{d x} [28 x^{3}] + \frac{d}{d x} [57 x^{2}] + \frac{d}{d x} [50 x] + \frac{d}{d x} [16]$ .

$\frac{d}{d x} [5 x^{4}] + \frac{d}{d x} [28 x^{3}] + \frac{d}{d x} [57 x^{2}] + \frac{d}{d x} [50 x] + \frac{d}{d x} [16]$

Step 1.2.2

Evaluate $\frac{d}{d x} [5 x^{4}]$ .

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Step 1.2.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{4}$ with respect to $x$ is $5 \frac{d}{d x} [x^{4}]$ .

$5 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [28 x^{3}] + \frac{d}{d x} [57 x^{2}] + \frac{d}{d x} [50 x] + \frac{d}{d x} [16]$

Step 1.2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$5 (4 x^{3}) + \frac{d}{d x} [28 x^{3}] + \frac{d}{d x} [57 x^{2}] + \frac{d}{d x} [50 x] + \frac{d}{d x} [16]$

Step 1.2.2.3

Multiply $4$ by $5$ .

$20 x^{3} + \frac{d}{d x} [28 x^{3}] + \frac{d}{d x} [57 x^{2}] + \frac{d}{d x} [50 x] + \frac{d}{d x} [16]$

Step 1.2.3

Evaluate $\frac{d}{d x} [28 x^{3}]$ .

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Step 1.2.3.1

Since $28$ is constant with respect to $x$ , the derivative of $28 x^{3}$ with respect to $x$ is $28 \frac{d}{d x} [x^{3}]$ .

$20 x^{3} + 28 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [57 x^{2}] + \frac{d}{d x} [50 x] + \frac{d}{d x} [16]$

Step 1.2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$20 x^{3} + 28 (3 x^{2}) + \frac{d}{d x} [57 x^{2}] + \frac{d}{d x} [50 x] + \frac{d}{d x} [16]$

Step 1.2.3.3

Multiply $3$ by $28$ .

$20 x^{3} + 84 x^{2} + \frac{d}{d x} [57 x^{2}] + \frac{d}{d x} [50 x] + \frac{d}{d x} [16]$

Step 1.2.4

Evaluate $\frac{d}{d x} [57 x^{2}]$ .

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Step 1.2.4.1

Since $57$ is constant with respect to $x$ , the derivative of $57 x^{2}$ with respect to $x$ is $57 \frac{d}{d x} [x^{2}]$ .

$20 x^{3} + 84 x^{2} + 57 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [50 x] + \frac{d}{d x} [16]$

Step 1.2.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$20 x^{3} + 84 x^{2} + 57 (2 x) + \frac{d}{d x} [50 x] + \frac{d}{d x} [16]$

Step 1.2.4.3

Multiply $2$ by $57$ .

$20 x^{3} + 84 x^{2} + 114 x + \frac{d}{d x} [50 x] + \frac{d}{d x} [16]$

Step 1.2.5

Evaluate $\frac{d}{d x} [50 x]$ .

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Step 1.2.5.1

Since $50$ is constant with respect to $x$ , the derivative of $50 x$ with respect to $x$ is $50 \frac{d}{d x} [x]$ .

$20 x^{3} + 84 x^{2} + 114 x + 50 \frac{d}{d x} [x] + \frac{d}{d x} [16]$

Step 1.2.5.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$20 x^{3} + 84 x^{2} + 114 x + 50 \cdot 1 + \frac{d}{d x} [16]$

Step 1.2.5.3

Multiply $50$ by $1$ .

$20 x^{3} + 84 x^{2} + 114 x + 50 + \frac{d}{d x} [16]$

Step 1.2.6

Differentiate using the Constant Rule.

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Step 1.2.6.1

Since $16$ is constant with respect to $x$ , the derivative of $16$ with respect to $x$ is $0$ .

$20 x^{3} + 84 x^{2} + 114 x + 50 + 0$

Step 1.2.6.2

Add $20 x^{3} + 84 x^{2} + 114 x + 50$ and $0$ .

$f'' (x) = 20 x^{3} + 84 x^{2} + 114 x + 50$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $20 x^{3} + 84 x^{2} + 114 x + 50$ .

$20 x^{3} + 84 x^{2} + 114 x + 50$

Step 2

Set the second derivative equal to $0$ then solve the equation $20 x^{3} + 84 x^{2} + 114 x + 50 = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$20 x^{3} + 84 x^{2} + 114 x + 50 = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $2$ out of $20 x^{3} + 84 x^{2} + 114 x + 50$ .

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Step 2.2.1.1

Factor $2$ out of $20 x^{3}$ .

$2 (10 x^{3}) + 84 x^{2} + 114 x + 50 = 0$

Step 2.2.1.2

Factor $2$ out of $84 x^{2}$ .

$2 (10 x^{3}) + 2 (42 x^{2}) + 114 x + 50 = 0$

Step 2.2.1.3

Factor $2$ out of $114 x$ .

$2 (10 x^{3}) + 2 (42 x^{2}) + 2 (57 x) + 50 = 0$

Step 2.2.1.4

Factor $2$ out of $50$ .

$2 (10 x^{3}) + 2 (42 x^{2}) + 2 (57 x) + 2 (25) = 0$

Step 2.2.1.5

Factor $2$ out of $2 (10 x^{3}) + 2 (42 x^{2})$ .

$2 (10 x^{3} + 42 x^{2}) + 2 (57 x) + 2 (25) = 0$

Step 2.2.1.6

Factor $2$ out of $2 (10 x^{3} + 42 x^{2}) + 2 (57 x)$ .

$2 (10 x^{3} + 42 x^{2} + 57 x) + 2 (25) = 0$

Step 2.2.1.7

Factor $2$ out of $2 (10 x^{3} + 42 x^{2} + 57 x) + 2 (25)$ .

$2 (10 x^{3} + 42 x^{2} + 57 x + 25) = 0$

Step 2.2.2

Factor.

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Step 2.2.2.1

Factor $10 x^{3} + 42 x^{2} + 57 x + 25$ using the rational roots test.

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Step 2.2.2.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 25, \pm 5$

$q = \pm 1, \pm 10, \pm 2, \pm 5$

Step 2.2.2.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 0.1, \pm 0.5, \pm 0.2, \pm 25, \pm 2.5, \pm 12.5, \pm 5$

Step 2.2.2.1.3

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

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Step 2.2.2.1.3.1

Substitute $- 1$ into the polynomial.

$10 {(- 1)}^{3} + 42 {(- 1)}^{2} + 57 \cdot - 1 + 25$

Step 2.2.2.1.3.2

Raise $- 1$ to the power of $3$ .

$10 \cdot - 1 + 42 {(- 1)}^{2} + 57 \cdot - 1 + 25$

Step 2.2.2.1.3.3

Multiply $10$ by $- 1$ .

$- 10 + 42 {(- 1)}^{2} + 57 \cdot - 1 + 25$

Step 2.2.2.1.3.4

Raise $- 1$ to the power of $2$ .

$- 10 + 42 \cdot 1 + 57 \cdot - 1 + 25$

Step 2.2.2.1.3.5

Multiply $42$ by $1$ .

$- 10 + 42 + 57 \cdot - 1 + 25$

Step 2.2.2.1.3.6

Add $- 10$ and $42$ .

$32 + 57 \cdot - 1 + 25$

Step 2.2.2.1.3.7

Multiply $57$ by $- 1$ .

$32 - 57 + 25$

Step 2.2.2.1.3.8

Subtract $57$ from $32$ .

$- 25 + 25$

Step 2.2.2.1.3.9

Add $- 25$ and $25$ .

$0$

Step 2.2.2.1.4

Since $- 1$ is a known root, divide the polynomial by $x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{10 x^{3} + 42 x^{2} + 57 x + 25}{x + 1}$

Step 2.2.2.1.5

Divide $10 x^{3} + 42 x^{2} + 57 x + 25$ by $x + 1$ .

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Step 2.2.2.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$1$

$10 x^{3}$

$42 x^{2}$

$57 x$

$25$

Step 2.2.2.1.5.2

Divide the highest order term in the dividend $10 x^{3}$ by the highest order term in divisor $x$ .

					$10 x^{2}$
	$x$	+	$1$		$10 x^{3}$	+	$42 x^{2}$	+	$57 x$	+	$25$

Step 2.2.2.1.5.3

Multiply the new quotient term by the divisor.

				$10 x^{2}$
$x$	+	$1$		$10 x^{3}$	+	$42 x^{2}$	+	$57 x$	+	$25$
			+	$10 x^{3}$	+	$10 x^{2}$

Step 2.2.2.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $10 x^{3} + 10 x^{2}$

				$10 x^{2}$
$x$	+	$1$		$10 x^{3}$	+	$42 x^{2}$	+	$57 x$	+	$25$
			-	$10 x^{3}$	-	$10 x^{2}$

Step 2.2.2.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$10 x^{2}$
$x$	+	$1$		$10 x^{3}$	+	$42 x^{2}$	+	$57 x$	+	$25$
			-	$10 x^{3}$	-	$10 x^{2}$
					+	$32 x^{2}$

Step 2.2.2.1.5.6

Pull the next terms from the original dividend down into the current dividend.

				$10 x^{2}$
$x$	+	$1$		$10 x^{3}$	+	$42 x^{2}$	+	$57 x$	+	$25$
			-	$10 x^{3}$	-	$10 x^{2}$
					+	$32 x^{2}$	+	$57 x$

Step 2.2.2.1.5.7

Divide the highest order term in the dividend $32 x^{2}$ by the highest order term in divisor $x$ .

				$10 x^{2}$	+	$32 x$
$x$	+	$1$		$10 x^{3}$	+	$42 x^{2}$	+	$57 x$	+	$25$
			-	$10 x^{3}$	-	$10 x^{2}$
					+	$32 x^{2}$	+	$57 x$

Step 2.2.2.1.5.8

Multiply the new quotient term by the divisor.

				$10 x^{2}$	+	$32 x$
$x$	+	$1$		$10 x^{3}$	+	$42 x^{2}$	+	$57 x$	+	$25$
			-	$10 x^{3}$	-	$10 x^{2}$
					+	$32 x^{2}$	+	$57 x$
					+	$32 x^{2}$	+	$32 x$

Step 2.2.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $32 x^{2} + 32 x$

				$10 x^{2}$	+	$32 x$
$x$	+	$1$		$10 x^{3}$	+	$42 x^{2}$	+	$57 x$	+	$25$
			-	$10 x^{3}$	-	$10 x^{2}$
					+	$32 x^{2}$	+	$57 x$
					-	$32 x^{2}$	-	$32 x$

Step 2.2.2.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$10 x^{2}$	+	$32 x$
$x$	+	$1$		$10 x^{3}$	+	$42 x^{2}$	+	$57 x$	+	$25$
			-	$10 x^{3}$	-	$10 x^{2}$
					+	$32 x^{2}$	+	$57 x$
					-	$32 x^{2}$	-	$32 x$
							+	$25 x$

Step 2.2.2.1.5.11

Pull the next terms from the original dividend down into the current dividend.

				$10 x^{2}$	+	$32 x$
$x$	+	$1$		$10 x^{3}$	+	$42 x^{2}$	+	$57 x$	+	$25$
			-	$10 x^{3}$	-	$10 x^{2}$
					+	$32 x^{2}$	+	$57 x$
					-	$32 x^{2}$	-	$32 x$
							+	$25 x$	+	$25$

Step 2.2.2.1.5.12

Divide the highest order term in the dividend $25 x$ by the highest order term in divisor $x$ .

				$10 x^{2}$	+	$32 x$	+	$25$
$x$	+	$1$		$10 x^{3}$	+	$42 x^{2}$	+	$57 x$	+	$25$
			-	$10 x^{3}$	-	$10 x^{2}$
					+	$32 x^{2}$	+	$57 x$
					-	$32 x^{2}$	-	$32 x$
							+	$25 x$	+	$25$

Step 2.2.2.1.5.13

Multiply the new quotient term by the divisor.

				$10 x^{2}$	+	$32 x$	+	$25$
$x$	+	$1$		$10 x^{3}$	+	$42 x^{2}$	+	$57 x$	+	$25$
			-	$10 x^{3}$	-	$10 x^{2}$
					+	$32 x^{2}$	+	$57 x$
					-	$32 x^{2}$	-	$32 x$
							+	$25 x$	+	$25$
							+	$25 x$	+	$25$

Step 2.2.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $25 x + 25$

				$10 x^{2}$	+	$32 x$	+	$25$
$x$	+	$1$		$10 x^{3}$	+	$42 x^{2}$	+	$57 x$	+	$25$
			-	$10 x^{3}$	-	$10 x^{2}$
					+	$32 x^{2}$	+	$57 x$
					-	$32 x^{2}$	-	$32 x$
							+	$25 x$	+	$25$
							-	$25 x$	-	$25$

Step 2.2.2.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$10 x^{2}$	+	$32 x$	+	$25$
$x$	+	$1$		$10 x^{3}$	+	$42 x^{2}$	+	$57 x$	+	$25$
			-	$10 x^{3}$	-	$10 x^{2}$
					+	$32 x^{2}$	+	$57 x$
					-	$32 x^{2}$	-	$32 x$
							+	$25 x$	+	$25$
							-	$25 x$	-	$25$
										$0$

Step 2.2.2.1.5.16

Since the remander is $0$ , the final answer is the quotient.

$10 x^{2} + 32 x + 25$

Step 2.2.2.1.6

Write $10 x^{3} + 42 x^{2} + 57 x + 25$ as a set of factors.

$2 ((x + 1) (10 x^{2} + 32 x + 25)) = 0$

Step 2.2.2.2

Remove unnecessary parentheses.

$2 (x + 1) (10 x^{2} + 32 x + 25) = 0$

Step 2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 1 = 0$

$10 x^{2} + 32 x + 25 = 0$

Step 2.4

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 2.4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.5

Set $10 x^{2} + 32 x + 25$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $10 x^{2} + 32 x + 25$ equal to $0$ .

$10 x^{2} + 32 x + 25 = 0$

Step 2.5.2

Solve $10 x^{2} + 32 x + 25 = 0$ for $x$ .

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Step 2.5.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.5.2.2

Substitute the values $a = 10$ , $b = 32$ , and $c = 25$ into the quadratic formula and solve for $x$ .

$\frac{- 32 \pm \sqrt{32^{2} - 4 \cdot (10 \cdot 25)}}{2 \cdot 10}$

Step 2.5.2.3

Simplify.

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Step 2.5.2.3.1

Simplify the numerator.

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Step 2.5.2.3.1.1

Raise $32$ to the power of $2$ .

$x = \frac{- 32 \pm \sqrt{1024 - 4 \cdot 10 \cdot 25}}{2 \cdot 10}$

Step 2.5.2.3.1.2

Multiply $- 4 \cdot 10 \cdot 25$ .

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Step 2.5.2.3.1.2.1

Multiply $- 4$ by $10$ .

$x = \frac{- 32 \pm \sqrt{1024 - 40 \cdot 25}}{2 \cdot 10}$

Step 2.5.2.3.1.2.2

Multiply $- 40$ by $25$ .

$x = \frac{- 32 \pm \sqrt{1024 - 1000}}{2 \cdot 10}$

Step 2.5.2.3.1.3

Subtract $1000$ from $1024$ .

$x = \frac{- 32 \pm \sqrt{24}}{2 \cdot 10}$

Step 2.5.2.3.1.4

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 2.5.2.3.1.4.1

Factor $4$ out of $24$ .

$x = \frac{- 32 \pm \sqrt{4 (6)}}{2 \cdot 10}$

Step 2.5.2.3.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 32 \pm \sqrt{2^{2} \cdot 6}}{2 \cdot 10}$

Step 2.5.2.3.1.5

Pull terms out from under the radical.

$x = \frac{- 32 \pm 2 \sqrt{6}}{2 \cdot 10}$

Step 2.5.2.3.2

Multiply $2$ by $10$ .

$x = \frac{- 32 \pm 2 \sqrt{6}}{20}$

Step 2.5.2.3.3

Simplify $\frac{- 32 \pm 2 \sqrt{6}}{20}$ .

$x = \frac{- 16 \pm \sqrt{6}}{10}$

Step 2.5.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.5.2.4.1

Simplify the numerator.

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Step 2.5.2.4.1.1

Raise $32$ to the power of $2$ .

$x = \frac{- 32 \pm \sqrt{1024 - 4 \cdot 10 \cdot 25}}{2 \cdot 10}$

Step 2.5.2.4.1.2

Multiply $- 4 \cdot 10 \cdot 25$ .

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Step 2.5.2.4.1.2.1

Multiply $- 4$ by $10$ .

$x = \frac{- 32 \pm \sqrt{1024 - 40 \cdot 25}}{2 \cdot 10}$

Step 2.5.2.4.1.2.2

Multiply $- 40$ by $25$ .

$x = \frac{- 32 \pm \sqrt{1024 - 1000}}{2 \cdot 10}$

Step 2.5.2.4.1.3

Subtract $1000$ from $1024$ .

$x = \frac{- 32 \pm \sqrt{24}}{2 \cdot 10}$

Step 2.5.2.4.1.4

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 2.5.2.4.1.4.1

Factor $4$ out of $24$ .

$x = \frac{- 32 \pm \sqrt{4 (6)}}{2 \cdot 10}$

Step 2.5.2.4.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 32 \pm \sqrt{2^{2} \cdot 6}}{2 \cdot 10}$

Step 2.5.2.4.1.5

Pull terms out from under the radical.

$x = \frac{- 32 \pm 2 \sqrt{6}}{2 \cdot 10}$

Step 2.5.2.4.2

Multiply $2$ by $10$ .

$x = \frac{- 32 \pm 2 \sqrt{6}}{20}$

Step 2.5.2.4.3

Simplify $\frac{- 32 \pm 2 \sqrt{6}}{20}$ .

$x = \frac{- 16 \pm \sqrt{6}}{10}$

Step 2.5.2.4.4

Change the $\pm$ to $+$ .

$x = \frac{- 16 + \sqrt{6}}{10}$

Step 2.5.2.4.5

Rewrite $- 16$ as $- 1 (16)$ .

$x = \frac{- 1 \cdot 16 + \sqrt{6}}{10}$

Step 2.5.2.4.6

Factor $- 1$ out of $\sqrt{6}$ .

$x = \frac{- 1 \cdot 16 - 1 (- \sqrt{6})}{10}$

Step 2.5.2.4.7

Factor $- 1$ out of $- 1 (16) - 1 (- \sqrt{6})$ .

$x = \frac{- 1 (16 - \sqrt{6})}{10}$

Step 2.5.2.4.8

Move the negative in front of the fraction.

$x = - \frac{16 - \sqrt{6}}{10}$

Step 2.5.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.5.2.5.1

Simplify the numerator.

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Step 2.5.2.5.1.1

Raise $32$ to the power of $2$ .

$x = \frac{- 32 \pm \sqrt{1024 - 4 \cdot 10 \cdot 25}}{2 \cdot 10}$

Step 2.5.2.5.1.2

Multiply $- 4 \cdot 10 \cdot 25$ .

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Step 2.5.2.5.1.2.1

Multiply $- 4$ by $10$ .

$x = \frac{- 32 \pm \sqrt{1024 - 40 \cdot 25}}{2 \cdot 10}$

Step 2.5.2.5.1.2.2

Multiply $- 40$ by $25$ .

$x = \frac{- 32 \pm \sqrt{1024 - 1000}}{2 \cdot 10}$

Step 2.5.2.5.1.3

Subtract $1000$ from $1024$ .

$x = \frac{- 32 \pm \sqrt{24}}{2 \cdot 10}$

Step 2.5.2.5.1.4

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 2.5.2.5.1.4.1

Factor $4$ out of $24$ .

$x = \frac{- 32 \pm \sqrt{4 (6)}}{2 \cdot 10}$

Step 2.5.2.5.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 32 \pm \sqrt{2^{2} \cdot 6}}{2 \cdot 10}$

Step 2.5.2.5.1.5

Pull terms out from under the radical.

$x = \frac{- 32 \pm 2 \sqrt{6}}{2 \cdot 10}$

Step 2.5.2.5.2

Multiply $2$ by $10$ .

$x = \frac{- 32 \pm 2 \sqrt{6}}{20}$

Step 2.5.2.5.3

Simplify $\frac{- 32 \pm 2 \sqrt{6}}{20}$ .

$x = \frac{- 16 \pm \sqrt{6}}{10}$

Step 2.5.2.5.4

Change the $\pm$ to $-$ .

$x = \frac{- 16 - \sqrt{6}}{10}$

Step 2.5.2.5.5

Rewrite $- 16$ as $- 1 (16)$ .

$x = \frac{- 1 \cdot 16 - \sqrt{6}}{10}$

Step 2.5.2.5.6

Factor $- 1$ out of $- \sqrt{6}$ .

$x = \frac{- 1 \cdot 16 - (\sqrt{6})}{10}$

Step 2.5.2.5.7

Factor $- 1$ out of $- 1 (16) - (\sqrt{6})$ .

$x = \frac{- 1 (16 + \sqrt{6})}{10}$

Step 2.5.2.5.8

Move the negative in front of the fraction.

$x = - \frac{16 + \sqrt{6}}{10}$

Step 2.5.2.6

The final answer is the combination of both solutions.

$x = - \frac{16 - \sqrt{6}}{10}, - \frac{16 + \sqrt{6}}{10}$

Step 2.6

The final solution is all the values that make $2 (x + 1) (10 x^{2} + 32 x + 25) = 0$ true.

$x = - 1, - \frac{16 - \sqrt{6}}{10}, - \frac{16 + \sqrt{6}}{10}$

Step 3

Find the points where the second derivative is $0$ .

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Step 3.1

Substitute $- 1$ in $f (x) = {(x + 1)}^{3} {(x + 2)}^{2}$ to find the value of $y$ .

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Step 3.1.1

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = {((- 1) + 1)}^{3} {((- 1) + 2)}^{2}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Add $- 1$ and $1$ .

$f (- 1) = 0^{3} {((- 1) + 2)}^{2}$

Step 3.1.2.2

Raising $0$ to any positive power yields $0$ .

$f (- 1) = 0 {((- 1) + 2)}^{2}$

Step 3.1.2.3

Add $- 1$ and $2$ .

$f (- 1) = 0 \cdot 1^{2}$

Step 3.1.2.4

One to any power is one.

$f (- 1) = 0 \cdot 1$

Step 3.1.2.5

Multiply $0$ by $1$ .

$f (- 1) = 0$

Step 3.1.2.6

The final answer is $0$ .

$0$

Step 3.2

The point found by substituting $- 1$ in $f (x) = {(x + 1)}^{3} {(x + 2)}^{2}$ is $(- 1, 0)$ . This point can be an inflection point.

$(- 1, 0)$

Step 3.3

Substitute $- 1.35505102$ in $f (x) = {(x + 1)}^{3} {(x + 2)}^{2}$ to find the value of $y$ .

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Step 3.3.1

Replace the variable $x$ with $- 1.35505102$ in the expression.

$f (- 1.35505102) = {((- 1.35505102) + 1)}^{3} {((- 1.35505102) + 2)}^{2}$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Add $- 1.35505102$ and $1$ .

$f (- 1.35505102) = {(- 0.35505102)}^{3} {((- 1.35505102) + 2)}^{2}$

Step 3.3.2.2

Raise $- 0.35505102$ to the power of $3$ .

$f (- 1.35505102) = - 0.04475816 {((- 1.35505102) + 2)}^{2}$

Step 3.3.2.3

Add $- 1.35505102$ and $2$ .

$f (- 1.35505102) = - 0.04475816 \cdot {0.64494897}^{2}$

Step 3.3.2.4

Raise $0.64494897$ to the power of $2$ .

$f (- 1.35505102) = - 0.04475816 \cdot 0.41595917$

Step 3.3.2.5

Multiply $- 0.04475816$ by $0.41595917$ .

$f (- 1.35505102) = - 0.01861757$

Step 3.3.2.6

The final answer is $- 0.01861757$ .

$- 0.01861757$

Step 3.4

The point found by substituting $- 1.35505102$ in $f (x) = {(x + 1)}^{3} {(x + 2)}^{2}$ is $(- 1.35505102, - 0.01861757)$ . This point can be an inflection point.

$(- 1.35505102, - 0.01861757)$

Step 3.5

Substitute $- 1.84494897$ in $f (x) = {(x + 1)}^{3} {(x + 2)}^{2}$ to find the value of $y$ .

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Step 3.5.1

Replace the variable $x$ with $- 1.84494897$ in the expression.

$f (- 1.84494897) = {((- 1.84494897) + 1)}^{3} {((- 1.84494897) + 2)}^{2}$

Step 3.5.2

Simplify the result.

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Step 3.5.2.1

Add $- 1.84494897$ and $1$ .

$f (- 1.84494897) = {(- 0.84494897)}^{3} {((- 1.84494897) + 2)}^{2}$

Step 3.5.2.2

Raise $- 0.84494897$ to the power of $3$ .

$f (- 1.84494897) = - 0.60324183 {((- 1.84494897) + 2)}^{2}$

Step 3.5.2.3

Add $- 1.84494897$ and $2$ .

$f (- 1.84494897) = - 0.60324183 \cdot {0.15505102}^{2}$

Step 3.5.2.4

Raise $0.15505102$ to the power of $2$ .

$f (- 1.84494897) = - 0.60324183 \cdot 0.02404082$

Step 3.5.2.5

Multiply $- 0.60324183$ by $0.02404082$ .

$f (- 1.84494897) = - 0.01450242$

Step 3.5.2.6

The final answer is $- 0.01450242$ .

$- 0.01450242$

Step 3.6

The point found by substituting $- 1.84494897$ in $f (x) = {(x + 1)}^{3} {(x + 2)}^{2}$ is $(- 1.84494897, - 0.01450242)$ . This point can be an inflection point.

$(- 1.84494897, - 0.01450242)$

Step 3.7

Determine the points that could be inflection points.

$(- 1, 0), (- 1.35505102, - 0.01861757), (- 1.84494897, - 0.01450242)$

Step 4

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

$(- \infty, - 1.84494897) \cup (- 1.84494897, - 1.35505102) \cup (- 1.35505102, - 1) \cup (- 1, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - 1.84494897)$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 1.94494897$ in the expression.

$f'' (- 1.94494897) = 20 {(- 1.94494897)}^{3} + 84 {(- 1.94494897)}^{2} + 114 (- 1.94494897) + 50$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Raise $- 1.94494897$ to the power of $3$ .

$f'' (- 1.94494897) = 20 \cdot - 7.35740454 + 84 {(- 1.94494897)}^{2} + 114 (- 1.94494897) + 50$

Step 5.2.1.2

Multiply $20$ by $- 7.35740454$ .

$f'' (- 1.94494897) = - 147.1480909 + 84 {(- 1.94494897)}^{2} + 114 (- 1.94494897) + 50$

Step 5.2.1.3

Raise $- 1.94494897$ to the power of $2$ .

$f'' (- 1.94494897) = - 147.1480909 + 84 \cdot 3.78282651 + 114 (- 1.94494897) + 50$

Step 5.2.1.4

Multiply $84$ by $3.78282651$ .

$f'' (- 1.94494897) = - 147.1480909 + 317.75742705 + 114 (- 1.94494897) + 50$

Step 5.2.1.5

Multiply $114$ by $- 1.94494897$ .

$f'' (- 1.94494897) = - 147.1480909 + 317.75742705 - 221.72418306 + 50$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Add $- 147.1480909$ and $317.75742705$ .

$f'' (- 1.94494897) = 170.60933614 - 221.72418306 + 50$

Step 5.2.2.2

Subtract $221.72418306$ from $170.60933614$ .

$f'' (- 1.94494897) = - 51.11484692 + 50$

Step 5.2.2.3

Add $- 51.11484692$ and $50$ .

$f'' (- 1.94494897) = - 1.11484692$

Step 5.2.3

The final answer is $- 1.11484692$ .

$- 1.11484692$

Step 5.3

At $- 1.94494897$ , the second derivative is $- 1.11484692$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, - 1.84494897)$

Decreasing on $(- \infty, - 1.84494897)$ since $f'' (x) < 0$

Step 6

Substitute a value from the interval $(- 1.84494897, - 1.35505102)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 1.6$ in the expression.

$f'' (- 1.6) = 20 {(- 1.6)}^{3} + 84 {(- 1.6)}^{2} + 114 (- 1.6) + 50$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Raise $- 1.6$ to the power of $3$ .

$f'' (- 1.6) = 20 \cdot - 4.096 + 84 {(- 1.6)}^{2} + 114 (- 1.6) + 50$

Step 6.2.1.2

Multiply $20$ by $- 4.096$ .

$f'' (- 1.6) = - 81.92 + 84 {(- 1.6)}^{2} + 114 (- 1.6) + 50$

Step 6.2.1.3

Raise $- 1.6$ to the power of $2$ .

$f'' (- 1.6) = - 81.92 + 84 \cdot 2.56 + 114 (- 1.6) + 50$

Step 6.2.1.4

Multiply $84$ by $2.56$ .

$f'' (- 1.6) = - 81.92 + 215.04 + 114 (- 1.6) + 50$

Step 6.2.1.5

Multiply $114$ by $- 1.6$ .

$f'' (- 1.6) = - 81.92 + 215.04 - 182.4 + 50$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Add $- 81.92$ and $215.04$ .

$f'' (- 1.6) = 133.12 - 182.4 + 50$

Step 6.2.2.2

Subtract $182.4$ from $133.12$ .

$f'' (- 1.6) = - 49.28 + 50$

Step 6.2.2.3

Add $- 49.28$ and $50$ .

$f'' (- 1.6) = 0.72$

Step 6.2.3

The final answer is $0.72$ .

$0.72$

Step 6.3

At $- 1.6$ , the second derivative is $0.72$ . Since this is positive, the second derivative is increasing on the interval $(- 1.84494897, - 1.35505102)$ .

Increasing on $(- 1.84494897, - 1.35505102)$ since $f'' (x) > 0$

Step 7

Substitute a value from the interval $(- 1.35505102, - 1)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- 1.17752551$ in the expression.

$f'' (- 1.17752551) = 20 {(- 1.17752551)}^{3} + 84 {(- 1.17752551)}^{2} + 114 (- 1.17752551) + 50$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Raise $- 1.17752551$ to the power of $3$ .

$f'' (- 1.17752551) = 20 \cdot - 1.63271723 + 84 {(- 1.17752551)}^{2} + 114 (- 1.17752551) + 50$

Step 7.2.1.2

Multiply $20$ by $- 1.63271723$ .

$f'' (- 1.17752551) = - 32.65434465 + 84 {(- 1.17752551)}^{2} + 114 (- 1.17752551) + 50$

Step 7.2.1.3

Raise $- 1.17752551$ to the power of $2$ .

$f'' (- 1.17752551) = - 32.65434465 + 84 \cdot 1.38656633 + 114 (- 1.17752551) + 50$

Step 7.2.1.4

Multiply $84$ by $1.38656633$ .

$f'' (- 1.17752551) = - 32.65434465 + 116.471572 + 114 (- 1.17752551) + 50$

Step 7.2.1.5

Multiply $114$ by $- 1.17752551$ .

$f'' (- 1.17752551) = - 32.65434465 + 116.471572 - 134.23790846 + 50$

Step 7.2.2

Simplify by adding and subtracting.

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Step 7.2.2.1

Add $- 32.65434465$ and $116.471572$ .

$f'' (- 1.17752551) = 83.81722735 - 134.23790846 + 50$

Step 7.2.2.2

Subtract $134.23790846$ from $83.81722735$ .

$f'' (- 1.17752551) = - 50.42068111 + 50$

Step 7.2.2.3

Add $- 50.42068111$ and $50$ .

$f'' (- 1.17752551) = - 0.42068111$

Step 7.2.3

The final answer is $- 0.42068111$ .

$- 0.42068111$

Step 7.3

At $- 1.17752551$ , the second derivative is $- 0.42068111$ . Since this is negative, the second derivative is decreasing on the interval $(- 1.35505102, - 1)$

Decreasing on $(- 1.35505102, - 1)$ since $f'' (x) < 0$

Step 8

Substitute a value from the interval $(- 1, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $- 0.9$ in the expression.

$f'' (- 0.9) = 20 {(- 0.9)}^{3} + 84 {(- 0.9)}^{2} + 114 (- 0.9) + 50$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Raise $- 0.9$ to the power of $3$ .

$f'' (- 0.9) = 20 \cdot - 0.729 + 84 {(- 0.9)}^{2} + 114 (- 0.9) + 50$

Step 8.2.1.2

Multiply $20$ by $- 0.729$ .

$f'' (- 0.9) = - 14.58 + 84 {(- 0.9)}^{2} + 114 (- 0.9) + 50$

Step 8.2.1.3

Raise $- 0.9$ to the power of $2$ .

$f'' (- 0.9) = - 14.58 + 84 \cdot 0.81 + 114 (- 0.9) + 50$

Step 8.2.1.4

Multiply $84$ by $0.81$ .

$f'' (- 0.9) = - 14.58 + 68.04 + 114 (- 0.9) + 50$

Step 8.2.1.5

Multiply $114$ by $- 0.9$ .

$f'' (- 0.9) = - 14.58 + 68.04 - 102.6 + 50$

Step 8.2.2

Simplify by adding and subtracting.

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Step 8.2.2.1

Add $- 14.58$ and $68.04$ .

$f'' (- 0.9) = 53.46 - 102.6 + 50$

Step 8.2.2.2

Subtract $102.6$ from $53.46$ .

$f'' (- 0.9) = - 49.14 + 50$

Step 8.2.2.3

Add $- 49.14$ and $50$ .

$f'' (- 0.9) = 0.86$

Step 8.2.3

The final answer is $0.86$ .

$0.86$

Step 8.3

At $- 0.9$ , the second derivative is $0.86$ . Since this is positive, the second derivative is increasing on the interval $(- 1, \infty)$ .

Increasing on $(- 1, \infty)$ since $f'' (x) > 0$

Step 9

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection points in this case are $(- 1.84494897, - 0.01450242), (- 1.35505102, - 0.01861757), (- 1, 0)$ .

$(- 1.84494897, - 0.01450242), (- 1.35505102, - 0.01861757), (- 1, 0)$

Step 10