Please help with a few problems I have before my final exam (:

Question

3. Let f(x)=S₀^x g(t)dt, 0<=x<=6, where g(t) is given by the graph below.

Note: I'm using "S" as the definite integral sign.

(a) Find the values x at which f(x) has a local extremum.

(b) Find the values x at which f(x) has an inflection point.

(c) Sketch the graph of f(x), 0<=x<=6.

Answer 1

Fortunately, you do not need the equation for g(t) to do this problem. Just remember that critical points of f(x) (local extrema and inflection points) occur when f'(x)=0. In one of your other problems you saw that f'(x)=g(x), so the critical points of f(x) occur when g(x)=0. Looks to me like that happens at 0, 2, and 6. I will leave it to you to classify these as local max/local min/inflection point.

Sketching the graph of f(x) is easy if you remember that f(x) is just the area between the graph of g(t) and the t-axis from 0 to x, with areas under the t-axis counting as negative. The area starts with 0 at 0, is increasingly negative until x=2, decreasingly negative between 2 and about 3.5, and positive thereafter.