Calculus Examples

$f (x) = - 2 x^{4} + 8 x^{3} - 12 x^{2}$

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

By the Sum Rule, the derivative of $- 2 x^{4} + 8 x^{3} - 12 x^{2}$ with respect to $x$ is $\frac{d}{d x} [- 2 x^{4}] + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [- 12 x^{2}]$ .

$\frac{d}{d x} [- 2 x^{4}] + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [- 12 x^{2}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [- 2 x^{4}]$ .

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Step 1.1.2.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x^{4}$ with respect to $x$ is $- 2 \frac{d}{d x} [x^{4}]$ .

$- 2 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [- 12 x^{2}]$

Step 1.1.2.2

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

$- 2 (4 x^{3}) + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [- 12 x^{2}]$

Step 1.1.2.3

Multiply $4$ by $- 2$ .

$- 8 x^{3} + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [- 12 x^{2}]$

Step 1.1.3

Evaluate $\frac{d}{d x} [8 x^{3}]$ .

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Step 1.1.3.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{3}$ with respect to $x$ is $8 \frac{d}{d x} [x^{3}]$ .

$- 8 x^{3} + 8 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 12 x^{2}]$

Step 1.1.3.2

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

$- 8 x^{3} + 8 (3 x^{2}) + \frac{d}{d x} [- 12 x^{2}]$

Step 1.1.3.3

Multiply $3$ by $8$ .

$- 8 x^{3} + 24 x^{2} + \frac{d}{d x} [- 12 x^{2}]$

Step 1.1.4

Evaluate $\frac{d}{d x} [- 12 x^{2}]$ .

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Step 1.1.4.1

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x^{2}$ with respect to $x$ is $- 12 \frac{d}{d x} [x^{2}]$ .

$- 8 x^{3} + 24 x^{2} - 12 \frac{d}{d x} [x^{2}]$

Step 1.1.4.2

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

$- 8 x^{3} + 24 x^{2} - 12 (2 x)$

Step 1.1.4.3

Multiply $2$ by $- 12$ .

$f' (x) = - 8 x^{3} + 24 x^{2} - 24 x$

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of $- 8 x^{3} + 24 x^{2} - 24 x$ with respect to $x$ is $\frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [24 x^{2}] + \frac{d}{d x} [- 24 x]$ .

$\frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [24 x^{2}] + \frac{d}{d x} [- 24 x]$

Step 1.2.2

Evaluate $\frac{d}{d x} [- 8 x^{3}]$ .

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Step 1.2.2.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x^{3}$ with respect to $x$ is $- 8 \frac{d}{d x} [x^{3}]$ .

$- 8 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [24 x^{2}] + \frac{d}{d x} [- 24 x]$

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 $n = 3$ .

$- 8 (3 x^{2}) + \frac{d}{d x} [24 x^{2}] + \frac{d}{d x} [- 24 x]$

Step 1.2.2.3

Multiply $3$ by $- 8$ .

$- 24 x^{2} + \frac{d}{d x} [24 x^{2}] + \frac{d}{d x} [- 24 x]$

Step 1.2.3

Evaluate $\frac{d}{d x} [24 x^{2}]$ .

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Step 1.2.3.1

Since $24$ is constant with respect to $x$ , the derivative of $24 x^{2}$ with respect to $x$ is $24 \frac{d}{d x} [x^{2}]$ .

$- 24 x^{2} + 24 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 24 x]$

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 $n = 2$ .

$- 24 x^{2} + 24 (2 x) + \frac{d}{d x} [- 24 x]$

Step 1.2.3.3

Multiply $2$ by $24$ .

$- 24 x^{2} + 48 x + \frac{d}{d x} [- 24 x]$

Step 1.2.4

Evaluate $\frac{d}{d x} [- 24 x]$ .

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Step 1.2.4.1

Since $- 24$ is constant with respect to $x$ , the derivative of $- 24 x$ with respect to $x$ is $- 24 \frac{d}{d x} [x]$ .

$- 24 x^{2} + 48 x - 24 \frac{d}{d x} [x]$

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 $n = 1$ .

$- 24 x^{2} + 48 x - 24 \cdot 1$

Step 1.2.4.3

Multiply $- 24$ by $1$ .

$f'' (x) = - 24 x^{2} + 48 x - 24$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $- 24 x^{2} + 48 x - 24$ .

$- 24 x^{2} + 48 x - 24$

Step 2

Set the second derivative equal to $0$ then solve the equation $- 24 x^{2} + 48 x - 24 = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$- 24 x^{2} + 48 x - 24 = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $- 24$ out of $- 24 x^{2} + 48 x - 24$ .

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Step 2.2.1.1

Factor $- 24$ out of $- 24 x^{2}$ .

$- 24 x^{2} + 48 x - 24 = 0$

Step 2.2.1.2

Factor $- 24$ out of $48 x$ .

$- 24 x^{2} - 24 (- 2 x) - 24 = 0$

Step 2.2.1.3

Factor $- 24$ out of $- 24$ .

$- 24 x^{2} - 24 (- 2 x) - 24 \cdot 1 = 0$

Step 2.2.1.4

Factor $- 24$ out of $- 24 (x^{2}) - 24 (- 2 x)$ .

$- 24 (x^{2} - 2 x) - 24 \cdot 1 = 0$

Step 2.2.1.5

Factor $- 24$ out of $- 24 (x^{2} - 2 x) - 24 (1)$ .

$- 24 (x^{2} - 2 x + 1) = 0$

Step 2.2.2

Factor using the perfect square rule.

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Step 2.2.2.1

Rewrite $1$ as $1^{2}$ .

$- 24 (x^{2} - 2 x + 1^{2}) = 0$

Step 2.2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 2.2.2.3

Rewrite the polynomial.

$- 24 (x^{2} - 2 \cdot x \cdot 1 + 1^{2}) = 0$

Step 2.2.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

$- 24 {(x - 1)}^{2} = 0$

Step 2.3

Divide each term in $- 24 {(x - 1)}^{2} = 0$ by $- 24$ and simplify.

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Step 2.3.1

Divide each term in $- 24 {(x - 1)}^{2} = 0$ by $- 24$ .

$\frac{- 24 {(x - 1)}^{2}}{- 24} = \frac{0}{- 24}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $- 24$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{- 24 {(x - 1)}^{2}}{- 24} = \frac{0}{- 24}$

Step 2.3.2.1.2

Divide ${(x - 1)}^{2}$ by $1$ .

${(x - 1)}^{2} = \frac{0}{- 24}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $0$ by $- 24$ .

${(x - 1)}^{2} = 0$

Step 2.4

Set the $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 2.5

Add $1$ to both sides of the equation.

$x = 1$

Step 3

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

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Step 3.1

Substitute $1$ in $f (x) = - 2 x^{4} + 8 x^{3} - 12 x^{2}$ to find the value of $y$ .

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Step 3.1.1

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

$f (1) = - 2 {(1)}^{4} + 8 {(1)}^{3} - 12 {(1)}^{2}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

One to any power is one.

$f (1) = - 2 \cdot 1 + 8 {(1)}^{3} - 12 {(1)}^{2}$

Step 3.1.2.1.2

Multiply $- 2$ by $1$ .

$f (1) = - 2 + 8 {(1)}^{3} - 12 {(1)}^{2}$

Step 3.1.2.1.3

One to any power is one.

$f (1) = - 2 + 8 \cdot 1 - 12 {(1)}^{2}$

Step 3.1.2.1.4

Multiply $8$ by $1$ .

$f (1) = - 2 + 8 - 12 {(1)}^{2}$

Step 3.1.2.1.5

One to any power is one.

$f (1) = - 2 + 8 - 12 \cdot 1$

Step 3.1.2.1.6

Multiply $- 12$ by $1$ .

$f (1) = - 2 + 8 - 12$

Step 3.1.2.2

Simplify by adding and subtracting.

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Step 3.1.2.2.1

Add $- 2$ and $8$ .

$f (1) = 6 - 12$

Step 3.1.2.2.2

Subtract $12$ from $6$ .

$f (1) = - 6$

Step 3.1.2.3

The final answer is $- 6$ .

$- 6$

Step 3.2

The point found by substituting $1$ in $f (x) = - 2 x^{4} + 8 x^{3} - 12 x^{2}$ is $(1, - 6)$ . This point can be an inflection point.

$(1, - 6)$

Step 4

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

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

Step 5

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

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Step 5.1

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

$f'' (0.9) = - 24 {(0.9)}^{2} + 48 (0.9) - 24$

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 $0.9$ to the power of $2$ .

$f'' (0.9) = - 24 \cdot 0.81 + 48 (0.9) - 24$

Step 5.2.1.2

Multiply $- 24$ by $0.81$ .

$f'' (0.9) = - 19.44 + 48 (0.9) - 24$

Step 5.2.1.3

Multiply $48$ by $0.9$ .

$f'' (0.9) = - 19.44 + 43.2 - 24$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Add $- 19.44$ and $43.2$ .

$f'' (0.9) = 23.76 - 24$

Step 5.2.2.2

Subtract $24$ from $23.76$ .

$f'' (0.9) = - 0.24$

Step 5.2.3

The final answer is $- 0.24$ .

$- 0.24$

Step 5.3

At $0.9$ , the second derivative is $- 0.24$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, 1)$

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

Step 6

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 6.1

Replace the variable $x$ with $1.1$ in the expression.

$f'' (1.1) = - 24 {(1.1)}^{2} + 48 (1.1) - 24$

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 $1.1$ to the power of $2$ .

$f'' (1.1) = - 24 \cdot 1.21 + 48 (1.1) - 24$

Step 6.2.1.2

Multiply $- 24$ by $1.21$ .

$f'' (1.1) = - 29.04 + 48 (1.1) - 24$

Step 6.2.1.3

Multiply $48$ by $1.1$ .

$f'' (1.1) = - 29.04 + 52.8 - 24$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Add $- 29.04$ and $52.8$ .

$f'' (1.1) = 23.76 - 24$

Step 6.2.2.2

Subtract $24$ from $23.76$ .

$f'' (1.1) = - 0.24$

Step 6.2.3

The final answer is $- 0.24$ .

$- 0.24$

Step 6.3

At $1.1$ , the second derivative is $- 0.24$ . Since this is negative, the second derivative is decreasing on the interval $(1, \infty)$

Decreasing on $(1, \infty)$ since $f'' (x) < 0$

Step 7

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. There are no points on the graph that satisfy these requirements.

No Inflection Points