Questions: Find the critical numbers of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) f(x)=5 x ln x x=

Solution Steps

To find the critical numbers of the function \( f(x) = 5x \ln x \), we need to find the derivative of the function and set it equal to zero. Critical numbers occur where the derivative is zero or undefined. We will solve for \( x \) to find these points.

We start with the function \( f(x) = 5x \ln x \). To find the critical numbers, we first compute the derivative of the function: \[ f'(x) = 5 \ln x + 5 \]

Next, we set the derivative equal to zero to find the critical points: \[ 5 \ln x + 5 = 0 \] This simplifies to: \[ \ln x = -1 \]

To solve for \( x \), we exponentiate both sides: \[ x = e^{-1} \] This can also be expressed as: \[ x = \frac{1}{e} \]

Final Answer