find the absolute and local maximum and minimum values of f.

Question

f(t) = 9 cos(t),    −3π/2 ≤ t ≤ 3π/2

absolute maximum value=

absolute minimum value=

local maximum value=

local minimum value=



Answer 1

f(t) = 9 cos(t),    −3π/2 ≤ t ≤ 3π/2

f ' (t) = -9 sin(t)

To get the local minimum and local maximum, we have to solve for (t,f(t)) when f ' (t) = 0.

-9 sin t = 0

sin t = 0

Within the interval −3π/2 ≤ t ≤ 3π/2, the following will be the values of t:

t = -π , 0, π 

If t = -π, then f(t) = -9

If t = 0 , then f(t) = 9

If t = π, then f(t) = -9

We can't have the values in (-∞, -2π]⋃[2π, ∞) because they are out of the given interval.

Therefore our local minima, as well as the absolute minima, are:

(-π, -9), (π, -9)

and our local maximum, as well as the absolute maximum, is:

(0, 9)