Find the Concavity f(x)=15x+15e^x

$f (x) = 15 x + 15 e^{x}$

Step 1

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

Find the second derivative.

Find the first derivative.

By the Sum Rule, the derivative of $15 x + 15 e^{x}$ with respect to $x$ is $\frac{d}{d x} [15 x] + \frac{d}{d x} [15 e^{x}]$ .

$f' (x) = \frac{d}{d x} (15 x) + \frac{d}{d x} (15 e^{x})$

Evaluate $\frac{d}{d x} [15 x]$ .

Since $15$ is constant with respect to $x$ , the derivative of $15 x$ with respect to $x$ is $15 \frac{d}{d x} [x]$ .

$f' (x) = 15 \frac{d}{d x} (x) + \frac{d}{d x} (15 e^{x})$

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

$f' (x) = 15 \cdot 1 + \frac{d}{d x} (15 e^{x})$

Multiply $15$ by $1$ .

$f' (x) = 15 + \frac{d}{d x} (15 e^{x})$

Evaluate $\frac{d}{d x} [15 e^{x}]$ .

Since $15$ is constant with respect to $x$ , the derivative of $15 e^{x}$ with respect to $x$ is $15 \frac{d}{d x} [e^{x}]$ .

$f' (x) = 15 + 15 \frac{d}{d x} (e^{x})$

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

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

Reorder terms.

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

Find the second derivative.

By the Sum Rule, the derivative of $15 e^{x} + 15$ with respect to $x$ is $\frac{d}{d x} [15 e^{x}] + \frac{d}{d x} [15]$ .

$f'' (x) = \frac{d}{d x} (15 e^{x}) + \frac{d}{d x} (15)$

Evaluate $\frac{d}{d x} [15 e^{x}]$ .

Since $15$ is constant with respect to $x$ , the derivative of $15 e^{x}$ with respect to $x$ is $15 \frac{d}{d x} [e^{x}]$ .

$f'' (x) = 15 \frac{d}{d x} (e^{x}) + \frac{d}{d x} (15)$

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

$f'' (x) = 15 e^{x} + \frac{d}{d x} (15)$

Differentiate using the Constant Rule.

Since $15$ is constant with respect to $x$ , the derivative of $15$ with respect to $x$ is $0$ .

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

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

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

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

$15 e^{x}$

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

Set the second derivative equal to $0$ .

$15 e^{x} = 0$

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

$\ln (15 e^{x}) = \ln (0)$

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

Undefined

There is no solution for $15 e^{x} = 0$

No solution

Step 2

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 3

The graph is concave up because the second derivative is positive.

The graph is concave up

Step 4

Find the Concavity f(x)=15x+15e^x

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