Given 6x squareroot3-12sinx +1 on the intervals [0,2pi) A)Find all critical numbers B)identify any local ma?

Question

B)identify any local maxium points of f on the interval [0,2pi)

Answer 1

I assume you mean
f(x) = 6x √3 - 12 sin x + 1
for the function. If not, the following will be wrong!

f'(x) = 6 √ 3 - 12 cos x
= 0 when cos x = √3 / 2.
This is true when x = π/6, 11π/6.

f"(x) = 12 sin x = 6 for x = π/6 and -6 for x = 11π/6. So the only local maximum is at (11π/6, 11π√3 + 7). (Depending on your terminology you may also need to include the endpoint at (0, 1).)