Question

Find the absolute maximum for the function f(x)=1/2x+cosx on [-pi/2, pi/2]

Answer 1

what your want to do here is find first derivation of f(x):

f(x) = 1/2x + cos x
f'(x) = 1/2 - sin x

now, you want to find critical points of your graph, and you do that by setting the first derivate to be equal to zero

0 = 1/2 - sin x
-1/2 = - sin x
sin x = 1/2
x = sin^-1 (1/2)
x = pi/6

we know that x-coordinate of maximum in this case must be greater then 0, because given interval is between [-pi/2, pi/2], and because of 1/2x in the original equation, if we insert a negative number, we can't reach our maximum.

with this, we can state that pi/6 is maximum on interval [-pi/2, pi/2]