Questions: List the critical numbers of the following function in increasing order. Enter N in any blank that you don't need to use. f(x)=5x e^(7x)

Transcript text: (5 points) List the critical numbers of the following function in increasing order. Enter $N$ in any blank that you don't need to use. \[ f(x)=5 x e^{7 x} \] $\square$ $\square$ $\square$

Solution

Solution Steps

To find the critical numbers of the function $ f(x) = 5x e^{7x} $, we need to follow these steps:

Compute the first derivative of the function.
Set the first derivative equal to zero and solve for $ x $.
Identify the critical numbers from the solutions obtained.

Step 1: Compute the First Derivative

We start with the function $ f(x) = 5x e^{7x} $. To find the critical numbers, we first compute the first derivative $ f'(x) $. Using the product rule, we find: \[ f'(x) = 35x e^{7x} + 5 e^{7x} \]

Step 2: Set the Derivative to Zero

Next, we set the first derivative equal to zero to find the critical points: \[ 35x e^{7x} + 5 e^{7x} = 0 \] Factoring out $ 5 e^{7x} $ gives us: \[ 5 e^{7x} (7x + 1) = 0 \] Since $ e^{7x} $ is never zero, we focus on the equation: \[ 7x + 1 = 0 \]

Step 3: Solve for $ x $

Solving for $ x $ yields: \[ 7x = -1 \quad \Rightarrow \quad x = -\frac{1}{7} \]