Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.


f(x) = x²e⁻ˣ

Verified step by step guidance

1. First, find the first derivative of the function f(x) = x²e⁻ˣ using the product rule, which states that if you have two functions u(x) and v(x), then the derivative is u'v + uv'.
2. Set the first derivative equal to zero to locate the critical points. This involves solving the equation f'(x) = 0.
3. Next, find the second derivative of the function f(x) to apply the Second Derivative Test. This may involve differentiating the first derivative you found in the previous step.
4. Evaluate the second derivative at each critical point found in step 2. This will help determine the concavity of the function at those points.
5. Finally, apply the Second Derivative Test: if f''(x) > 0 at a critical point, it is a local minimum; if f''(x) < 0, it is a local maximum; if f''(x) = 0, the test is inconclusive.