need help with B Thanks The graph of the derivative f of a f

need help with B. Thanks!

The graph of the derivative f\' of a function f is shown. On what intervals is f increasing or decreasing? (Enter your answers using interval notation.) At what values of x does f have a local maximum or minimum? (Enter your answers as a comma-separate If an answer does not exist, enter ONE.)

Solution

the given graph is parbola

the standard equation of parabola is y = -a(x-h)^2 +k

where (h,k) =(2,2)

y = -a(x-2)^2 +2

it is passing through (1,0)

0 = -a(1-2)^2 +2

-2 = -a

a=2

y = 2(x-2)^2 +2

this is f\'(x)

f\'(x) = 2(x^2 -4x+4) +2

f\'(x) = 2x^2 -8x +8+2

f\'(x) = 2x^2 -8x+10

now integrate this

f(x) = 2x^3 /3 -8/2 x^2 +10x

f(x) = 2/3 x^3 4x^2 +10x

now find f\'\'(x) = 4x - 8

now for local maximum f\'\'(x) shuold be negative

now for local minimum f\'\'(x) shuold be positive

(-2, -1) , (5, -2.5) f has local maximum

local minimum (1,1) (5,1)