Calculus Examples

$(x^{2} - 12 x + 37) e^{x}$

Step 1

Write $(x^{2} - 12 x + 37) e^{x}$ as a function.

$f (x) = (x^{2} - 12 x + 37) e^{x}$

Step 2

Find the $x$ values where the second derivative is equal to $0$ .

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Step 2.1

Find the second derivative.

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Step 2.1.1

Find the first derivative.

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Step 2.1.1.1

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 $f (x) = x^{2} - 12 x + 37$ and $g (x) = e^{x}$ .

$(x^{2} - 12 x + 37) \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{2} - 12 x + 37]$

Step 2.1.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$(x^{2} - 12 x + 37) e^{x} + e^{x} \frac{d}{d x} [x^{2} - 12 x + 37]$

Step 2.1.1.3

Differentiate.

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Step 2.1.1.3.1

By the Sum Rule, the derivative of $x^{2} - 12 x + 37$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [37]$ .

$(x^{2} - 12 x + 37) e^{x} + e^{x} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [37])$

Step 2.1.1.3.2

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

$(x^{2} - 12 x + 37) e^{x} + e^{x} (2 x + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [37])$

Step 2.1.1.3.3

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

$(x^{2} - 12 x + 37) e^{x} + e^{x} (2 x - 12 \frac{d}{d x} [x] + \frac{d}{d x} [37])$

Step 2.1.1.3.4

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

$(x^{2} - 12 x + 37) e^{x} + e^{x} (2 x - 12 \cdot 1 + \frac{d}{d x} [37])$

Step 2.1.1.3.5

Multiply $- 12$ by $1$ .

$(x^{2} - 12 x + 37) e^{x} + e^{x} (2 x - 12 + \frac{d}{d x} [37])$

Step 2.1.1.3.6

Since $37$ is constant with respect to $x$ , the derivative of $37$ with respect to $x$ is $0$ .

$(x^{2} - 12 x + 37) e^{x} + e^{x} (2 x - 12 + 0)$

Step 2.1.1.3.7

Add $2 x - 12$ and $0$ .

$(x^{2} - 12 x + 37) e^{x} + e^{x} (2 x - 12)$

Step 2.1.1.4

Simplify.

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Step 2.1.1.4.1

Apply the distributive property.

$x^{2} e^{x} - 12 x e^{x} + 37 e^{x} + e^{x} (2 x - 12)$

Step 2.1.1.4.2

Apply the distributive property.

$x^{2} e^{x} - 12 x e^{x} + 37 e^{x} + e^{x} (2 x) + e^{x} \cdot - 12$

Step 2.1.1.4.3

Combine terms.

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Step 2.1.1.4.3.1

Move $- 12$ to the left of $e^{x}$ .

$x^{2} e^{x} - 12 x e^{x} + 37 e^{x} + e^{x} (2 x) - 12 \cdot e^{x}$

Step 2.1.1.4.3.2

Add $- 12 x e^{x}$ and $e^{x} (2 x)$ .

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Step 2.1.1.4.3.2.1

Move $e^{x}$ .

$x^{2} e^{x} - 12 x e^{x} + 2 x e^{x} + 37 e^{x} - 12 e^{x}$

Step 2.1.1.4.3.2.2

Add $- 12 x e^{x}$ and $2 x e^{x}$ .

$x^{2} e^{x} - 10 x e^{x} + 37 e^{x} - 12 e^{x}$

Step 2.1.1.4.3.3

Subtract $12 e^{x}$ from $37 e^{x}$ .

$x^{2} e^{x} - 10 x e^{x} + 25 e^{x}$

Step 2.1.1.4.4

Reorder terms.

$e^{x} x^{2} - 10 e^{x} x + 25 e^{x}$

Step 2.1.1.4.5

Reorder factors in $e^{x} x^{2} - 10 e^{x} x + 25 e^{x}$ .

$f' (x) = x^{2} e^{x} - 10 x e^{x} + 25 e^{x}$

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

By the Sum Rule, the derivative of $x^{2} e^{x} - 10 x e^{x} + 25 e^{x}$ with respect to $x$ is $\frac{d}{d x} [x^{2} e^{x}] + \frac{d}{d x} [- 10 x e^{x}] + \frac{d}{d x} [25 e^{x}]$ .

$\frac{d}{d x} [x^{2} e^{x}] + \frac{d}{d x} [- 10 x e^{x}] + \frac{d}{d x} [25 e^{x}]$

Step 2.1.2.2

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

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Step 2.1.2.2.1

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 $f (x) = x^{2}$ and $g (x) = e^{x}$ .

$x^{2} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 10 x e^{x}] + \frac{d}{d x} [25 e^{x}]$

Step 2.1.2.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$x^{2} e^{x} + e^{x} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 10 x e^{x}] + \frac{d}{d x} [25 e^{x}]$

Step 2.1.2.2.3

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

$x^{2} e^{x} + e^{x} (2 x) + \frac{d}{d x} [- 10 x e^{x}] + \frac{d}{d x} [25 e^{x}]$

Step 2.1.2.3

Evaluate $\frac{d}{d x} [- 10 x e^{x}]$ .

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Step 2.1.2.3.1

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10 x e^{x}$ with respect to $x$ is $- 10 \frac{d}{d x} [x e^{x}]$ .

$x^{2} e^{x} + e^{x} (2 x) - 10 \frac{d}{d x} [x e^{x}] + \frac{d}{d x} [25 e^{x}]$

Step 2.1.2.3.2

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 $f (x) = x$ and $g (x) = e^{x}$ .

$x^{2} e^{x} + e^{x} (2 x) - 10 (x \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x]) + \frac{d}{d x} [25 e^{x}]$

Step 2.1.2.3.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$x^{2} e^{x} + e^{x} (2 x) - 10 (x e^{x} + e^{x} \frac{d}{d x} [x]) + \frac{d}{d x} [25 e^{x}]$

Step 2.1.2.3.4

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

$x^{2} e^{x} + e^{x} (2 x) - 10 (x e^{x} + e^{x} \cdot 1) + \frac{d}{d x} [25 e^{x}]$

Step 2.1.2.3.5

Multiply $e^{x}$ by $1$ .

$x^{2} e^{x} + e^{x} (2 x) - 10 (x e^{x} + e^{x}) + \frac{d}{d x} [25 e^{x}]$

Step 2.1.2.4

Evaluate $\frac{d}{d x} [25 e^{x}]$ .

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Step 2.1.2.4.1

Since $25$ is constant with respect to $x$ , the derivative of $25 e^{x}$ with respect to $x$ is $25 \frac{d}{d x} [e^{x}]$ .

$x^{2} e^{x} + e^{x} (2 x) - 10 (x e^{x} + e^{x}) + 25 \frac{d}{d x} [e^{x}]$

Step 2.1.2.4.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$x^{2} e^{x} + e^{x} (2 x) - 10 (x e^{x} + e^{x}) + 25 e^{x}$

Step 2.1.2.5

Simplify.

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Step 2.1.2.5.1

Apply the distributive property.

$x^{2} e^{x} + e^{x} (2 x) - 10 (x e^{x}) - 10 e^{x} + 25 e^{x}$

Step 2.1.2.5.2

Combine terms.

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Step 2.1.2.5.2.1

Subtract $10 x e^{x}$ from $e^{x} (2 x)$ .

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Step 2.1.2.5.2.1.1

Move $e^{x}$ .

$x^{2} e^{x} + 2 x e^{x} - 10 x e^{x} - 10 e^{x} + 25 e^{x}$

Step 2.1.2.5.2.1.2

Subtract $10 x e^{x}$ from $2 x e^{x}$ .

$x^{2} e^{x} - 8 x e^{x} - 10 e^{x} + 25 e^{x}$

Step 2.1.2.5.2.2

Add $- 10 e^{x}$ and $25 e^{x}$ .

$x^{2} e^{x} - 8 x e^{x} + 15 e^{x}$

Step 2.1.2.5.3

Reorder terms.

$e^{x} x^{2} - 8 e^{x} x + 15 e^{x}$

Step 2.1.2.5.4

Reorder factors in $e^{x} x^{2} - 8 e^{x} x + 15 e^{x}$ .

$f'' (x) = x^{2} e^{x} - 8 x e^{x} + 15 e^{x}$

Step 2.1.3

The second derivative of $f (x)$ with respect to $x$ is $x^{2} e^{x} - 8 x e^{x} + 15 e^{x}$ .

$x^{2} e^{x} - 8 x e^{x} + 15 e^{x}$

Step 2.2

Set the second derivative equal to $0$ then solve the equation $x^{2} e^{x} - 8 x e^{x} + 15 e^{x} = 0$ .

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Step 2.2.1

Set the second derivative equal to $0$ .

$x^{2} e^{x} - 8 x e^{x} + 15 e^{x} = 0$

Step 2.2.2

Factor the left side of the equation.

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Step 2.2.2.1

Factor $e^{x}$ out of $x^{2} e^{x} - 8 x e^{x} + 15 e^{x}$ .

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Step 2.2.2.1.1

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

$e^{x} x^{2} - 8 x e^{x} + 15 e^{x} = 0$

Step 2.2.2.1.2

Factor $e^{x}$ out of $- 8 x e^{x}$ .

$e^{x} x^{2} + e^{x} (- 8 x) + 15 e^{x} = 0$

Step 2.2.2.1.3

Factor $e^{x}$ out of $15 e^{x}$ .

$e^{x} x^{2} + e^{x} (- 8 x) + e^{x} \cdot 15 = 0$

Step 2.2.2.1.4

Factor $e^{x}$ out of $e^{x} x^{2} + e^{x} (- 8 x)$ .

$e^{x} (x^{2} - 8 x) + e^{x} \cdot 15 = 0$

Step 2.2.2.1.5

Factor $e^{x}$ out of $e^{x} (x^{2} - 8 x) + e^{x} \cdot 15$ .

$e^{x} (x^{2} - 8 x + 15) = 0$

Step 2.2.2.2

Factor.

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Step 2.2.2.2.1

Factor $x^{2} - 8 x + 15$ using the AC method.

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Step 2.2.2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $15$ and whose sum is $- 8$ .

$- 5, - 3$

Step 2.2.2.2.1.2

Write the factored form using these integers.

$e^{x} ((x - 5) (x - 3)) = 0$

Step 2.2.2.2.2

Remove unnecessary parentheses.

$e^{x} (x - 5) (x - 3) = 0$

Step 2.2.3

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

$e^{x} = 0$

$x - 5 = 0$

$x - 3 = 0$

Step 2.2.4

Set $e^{x}$ equal to $0$ and solve for $x$ .

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Step 2.2.4.1

Set $e^{x}$ equal to $0$ .

$e^{x} = 0$

Step 2.2.4.2

Solve $e^{x} = 0$ for $x$ .

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Step 2.2.4.2.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{x}) = \ln (0)$

Step 2.2.4.2.2

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

Step 2.2.4.2.3

There is no solution for $e^{x} = 0$

No solution

Step 2.2.5

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 2.2.5.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 2.2.5.2

Add $5$ to both sides of the equation.

$x = 5$

Step 2.2.6

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 2.2.6.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 2.2.6.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.2.7

The final solution is all the values that make $e^{x} (x - 5) (x - 3) = 0$ true.

$x = 5, 3$

Step 3

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4

Create intervals around the $x$ -values where the second derivative is zero or undefined.

$(- \infty, 3) \cup (3, 5) \cup (5, \infty)$

Step 5

Substitute any number from the interval $(- \infty, 3)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

Replace the variable $x$ with $0$ in the expression.

$f'' (0) = {(0)}^{2} e^{0} - 8 \cdot 0 \cdot e^{0} + 15 e^{0}$

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

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

$f'' (0) = 0 e^{0} - 8 \cdot 0 \cdot e^{0} + 15 e^{0}$

Step 5.2.1.2

Anything raised to $0$ is $1$ .

$f'' (0) = 0 \cdot 1 - 8 \cdot 0 \cdot e^{0} + 15 e^{0}$

Step 5.2.1.3

Multiply $0$ by $1$ .

$f'' (0) = 0 - 8 \cdot 0 \cdot e^{0} + 15 e^{0}$

Step 5.2.1.4

Multiply $- 8$ by $0$ .

$f'' (0) = 0 + 0 \cdot e^{0} + 15 e^{0}$

Step 5.2.1.5

Anything raised to $0$ is $1$ .

$f'' (0) = 0 + 0 \cdot 1 + 15 e^{0}$

Step 5.2.1.6

Multiply $0$ by $1$ .

$f'' (0) = 0 + 0 + 15 e^{0}$

Step 5.2.1.7

Anything raised to $0$ is $1$ .

$f'' (0) = 0 + 0 + 15 \cdot 1$

Step 5.2.1.8

Multiply $15$ by $1$ .

$f'' (0) = 0 + 0 + 15$

Step 5.2.2

Simplify by adding numbers.

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Step 5.2.2.1

Add $0$ and $0$ .

$f'' (0) = 0 + 15$

Step 5.2.2.2

Add $0$ and $15$ .

$f'' (0) = 15$

Step 5.2.3

The final answer is $15$ .

$15$

Step 5.3

The graph is concave up on the interval $(- \infty, 3)$ because $f'' (0)$ is positive.

Concave up on $(- \infty, 3)$ since $f'' (x)$ is positive

Step 6

Substitute any number from the interval $(3, 5)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

Replace the variable $x$ with $4$ in the expression.

$f'' (4) = {(4)}^{2} e^{4} - 8 \cdot 4 \cdot e^{4} + 15 e^{4}$

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 $4$ to the power of $2$ .

$f'' (4) = 16 e^{4} - 8 \cdot 4 \cdot e^{4} + 15 e^{4}$

Step 6.2.1.2

Multiply $- 8$ by $4$ .

$f'' (4) = 16 e^{4} - 32 e^{4} + 15 e^{4}$

Step 6.2.2

Simplify by adding terms.

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Step 6.2.2.1

Subtract $32 e^{4}$ from $16 e^{4}$ .

$f'' (4) = - 16 e^{4} + 15 e^{4}$

Step 6.2.2.2

Add $- 16 e^{4}$ and $15 e^{4}$ .

$f'' (4) = - e^{4}$

Step 6.2.3

The final answer is $- e^{4}$ .

$- e^{4}$

Step 6.3

The graph is concave down on the interval $(3, 5)$ because $f'' (4)$ is negative.

Concave down on $(3, 5)$ since $f'' (x)$ is negative

Step 7

Substitute any number from the interval $(5, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 7.1

Replace the variable $x$ with $8$ in the expression.

$f'' (8) = {(8)}^{2} e^{8} - 8 \cdot 8 \cdot e^{8} + 15 e^{8}$

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 $8$ to the power of $2$ .

$f'' (8) = 64 e^{8} - 8 \cdot 8 \cdot e^{8} + 15 e^{8}$

Step 7.2.1.2

Multiply $- 8$ by $8$ .

$f'' (8) = 64 e^{8} - 64 e^{8} + 15 e^{8}$

Step 7.2.2

Simplify by adding terms.

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Step 7.2.2.1

Subtract $64 e^{8}$ from $64 e^{8}$ .

$f'' (8) = 0 + 15 e^{8}$

Step 7.2.2.2

Add $0$ and $15 e^{8}$ .

$f'' (8) = 15 e^{8}$

Step 7.2.3

The final answer is $15 e^{8}$ .

$15 e^{8}$

Step 7.3

The graph is concave up on the interval $(5, \infty)$ because $f'' (8)$ is positive.

Concave up on $(5, \infty)$ since $f'' (x)$ is positive

Step 8

The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.

Concave up on $(- \infty, 3)$ since $f'' (x)$ is positive

Concave down on $(3, 5)$ since $f'' (x)$ is negative

Concave up on $(5, \infty)$ since $f'' (x)$ is positive

Step 9