characteristics of a graph

Question

Below is the graph of the function . If 

1. For what value(s) of x will g(x) have an extremum? Identify maximum or minimum.

2. For what value(s) of x will g(x) have a point of inflection?

3. Where is g(x) concave down?

4. Evaluate g(0), g(–4), and g(2).

5. Is  ?

Answer 1

Question 1: g(x) is the area under the curve of f(t). Notice the f(t) has negative area between -4 and 0 and that area keeps getting larger (more negative) the closer you get to zero until the area is as large a negative value as it's going to get when you get to zero because after that, you start adding positive area to it so it starts getting closer to zero. So at x = 0, g(x) has a relative minimum.

Question 2: g(x) will have a point of inflection when g''(x) changes sign. Since g(x) = ∫f(t) dt then g'(x) = f(t) and g''(x) = f '(t). That means to find points of inflections of g(x), we need to look at where the sign changes are on f'(t). On (-4, 0), the slope of f(t) is positive. On (0, 2) the slope is also positive but at x = 2, the slope changes to negative. So x = 2 is a point of inflection of g(x). Continuing, on (2, 4) the slope of f(t) is negative then on (4, 6), it's positive so at x = 4, there is another point of inflection for g(x).

Question 3. g(x) is concave down when g''(x) is negative meaning when f '(t) is negative. As mentioned above that is on (2, 4)

Question 4) g(0) is the area between -4 and 0 (Area of a triangle = 1/2bh)

g(-4) = 0 since there is no area between x = -4 and x = -4

g(2) add g(0) to 1/4 of the area of the circle of radius 2

Question 5) The equation for the circle shown is (x - 2)² + y² = 4 meaning the upper portion of the circle is given by y = √(4 - (x-2)²) so you can use your calculator to do ₀∫¹(4 - (x-2)²)^0.5dx and add it to the area you got of the triangle to the left of zero.