Let f(x) = (3+2x-x^2)^2 be defined for the interval [-2,3]. If M is the y-coordinate of the absolute maximum and m is the y-coordinate of the absolute minimum of f(x) on the interval, what is the value of (M+m)?

Question

Let f(x) = (3+2x-x^2)^2 be defined for the interval [-2,3].  If M is the y-coordinate of the absolute maximum and m is the y-coordinate of the absolute minimum of f(x) on the interval, what is the value of (M+m)?

Reiny · Accepted Answer

f(x) = (3+2x-x^2)^2
= (3 - x)^2 (1 + x)^2
so there are double roots at x = -1 and x = 3

https://www.wolframalpha.com/input/?i=plot+f%28x%29+%3D+%283%2B2x-x%5E2%29%5E2

f(-2) = 5^2(-1)^2 = 25
f(3) = 0

my graph shows a local max at x = appr 1, but is is lower than (-2,25)

so M = 25 and m = 0

continue