use the second derivative test for local extrema to find the x - values of all local extrema for the given…

use the second derivative test for local extrema to find the x - values of all local extrema for the given function.\nf(x)=-11x + 22\\cos x on 0,2\\pi\n(type exact answers, using \\pi as needed. simplify your answers.)\nd. the function f(x) has no local minimums.\ndoes the function have any local maximums? select the correct choice, and, if necessary, fill in the answer box(es) to complete your choice.\na. the function f(x) has one local maximum, located at x=\n(type an exact answer, using \\pi as needed. simplify your answer.)\nb. the function f(x) has two local maximums, located at x= and x=\n(type exact answers, using \\pi as needed. simplify your answers.)\nc. the function f(x) has three local maximums, located at x=, x=, and x=\n(type exact answers, using \\pi as needed. simplify your answers.)\nd. the function f(x) has no local maximums.

use the second derivative test for local extrema to find the x - values of all local extrema for the given function.\nf(x)=-11x + 22\\cos x on 0,2\\pi\n(type exact answers, using \\pi as needed. simplify your answers.)\nd. the function f(x) has no local minimums.\ndoes the function have any local maximums? select the correct choice, and, if necessary, fill in the answer box(es) to complete your choice.\na. the function f(x) has one local maximum, located at x=\n(type an exact answer, using \\pi as needed. simplify your answer.)\nb. the function f(x) has two local maximums, located at x= and x=\n(type exact answers, using \\pi as needed. simplify your answers.)\nc. the function f(x) has three local maximums, located at x=, x=, and x=\n(type exact answers, using \\pi as needed. simplify your answers.)\nd. the function f(x) has no local maximums.

Answer

Explanation:

Step1: Find the first derivative

The derivative of (y = - 11x+22\cos x) is (y^\prime=f^\prime(x)=-11 - 22\sin x). Set (y^\prime = 0), then (-11-22\sin x=0), which simplifies to (\sin x=-\frac{1}{2}). On the interval ([0,2\pi]), (x = \frac{7\pi}{6}) or (x=\frac{11\pi}{6}).

Step2: Find the second derivative

The second - derivative (y^{\prime\prime}=f^{\prime\prime}(x)=-22\cos x).

  • When (x = \frac{7\pi}{6}), (y^{\prime\prime}\left(\frac{7\pi}{6}\right)=-22\cos\left(\frac{7\pi}{6}\right)=-22\times\left(-\frac{\sqrt{3}}{2}\right)=11\sqrt{3}>0). So (x = \frac{7\pi}{6}) is a local minimum.
  • When (x=\frac{11\pi}{6}), (y^{\prime\prime}\left(\frac{11\pi}{6}\right)=-22\cos\left(\frac{11\pi}{6}\right)=-22\times\frac{\sqrt{3}}{2}=-11\sqrt{3}<0). So (x=\frac{11\pi}{6}) is a local maximum.

Answer:

A. The function (f(x)) has one local maximum, located at (x = \frac{11\pi}{6})