Questions: A function f has derivative f'(x)=x(x+3) e^x, and a critical point at x=0. Classify the critical point. LOCAL MAXIMUM LOCAL MINIMUM NOT A LOCAL EXTREMUM

Solution Steps

To classify the critical point at $x = 0$, we need to analyze the second derivative $f''(x)$. If $f''(0) > 0$, the critical point is a local minimum. If $f''(0) < 0$, the critical point is a local maximum. If $f''(0) = 0$, the test is inconclusive.

The first derivative of the function is given by

$$f'(x) = x(x + 3)e^x.$$

To classify the critical point at $x = 0$, we need to compute the second derivative $f''(x)$. The second derivative is calculated as follows:

$$f''(x) = f'(x) + \text{additional terms from the product rule}.$$

After simplification, we find that

$$f''(0) = 3.$$

To classify the critical point at $x = 0$, we evaluate $f''(0)$:

• If $f''(0) > 0$, then $x = 0$ is a local minimum.
• If $f''(0) < 0$, then $x = 0$ is a local maximum.
• If $f''(0) = 0$, the test is inconclusive.

Since $f''(0) = 3 > 0$, we conclude that $x = 0$ is a local minimum.

Final Answer