Question

Find the maximum and minimum values of f(x)=x^3+3x^2 on [-3,2]

Answer 1

to find maximums and minimums, we take the derivative and set it to zero

f'(x)=3x^2+6x
0=3x^2+6x
0=x^2+2x
x=0 or x=-2
These points are critical points, but we have to figure out if it is a maximum or minimum. The point (0,0) is surrounded by the points (-1,2) and (1,4), making this point a minimum. The point (-2,4) is surrounded by the points (-1,2) and (-3,0), making this point a maximum.

We must also test the endpoints. The point (-3,0) is an endpoint, as is (2,20). Now we know that there is one maximum, (2,20) and two minimums (-3,0) and (0,0).