Advanced Calculus Limits And Integration Exercises

This question requires utilizing knowledge of limits, continuity, differentiation, integration, optimization, and graph sketching.

Let f(x) = (x-1)(x-2)(x-3)e^x on the interval [0,5].

a) Find all critical points of f(x) on the interval. Classify each critical point as a local minimum, local maximum, or neither.

b) Use the First Derivative Test to determine all intervals on which f(x) is increasing or decreasing.

c) Find all inflection points of f(x) on the interval.

d) Use the Second Derivative Test to determine all intervals on which f(x) is concave up or concave down.