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.\n\nf(x)= - 11x + 22cosx on 0,2π\n\ndoes the function have any local minimums? select the correct choice, and, if necessary, fill in the answer box(es) to complete your choice.\n\na. the function f(x) has one local minimum, located at x=\n(type an exact answer, using π as needed. simplify your answer.)\nb. the function f(x) has two local minimums, located at x= and x=\n(type exact answers, using π as needed. simplify your answers)\nc. the function f(x) has three local minimums, located at x=,x=, and x=\n(type exact answers, using π as needed. simplify your answers.)\nd. the function f(x) has no local minimums.

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

Answer

Explanation:

Step1: Find the first derivative

The derivative of (y = - 11x+22\cos x) is (y^\prime=-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 of (y=-11x + 22\cos x) is (y^{\prime\prime}=-22\cos x).

Step3: Evaluate the second derivative at critical points

  • 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).
  • 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).

Answer:

A. The function (f(x)) has one local minimum, located at (x = \frac{7\pi}{6})