determine the location and value of the absolute extreme values of f on the given interval, if they…

determine the location and value of the absolute extreme values of f on the given interval, if they exist.\n\n$f(x)=\\cos 3x$ on $\\left-\\frac{\\pi}{4},\\frac{\\pi}{2}\\right$\n\nwhat is/are the absolute maximum/maxima of f on the given interval? select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.\n\na. the absolute maximum/maxima is/are 1 at $x = 0$\n\n(use a comma to separate answers as needed. type an exact answer, using $\\pi$ as needed)\n\nb. there is no absolute maximum of f on the given interval\n\nwhat is/are the absolute minimum/minima of f on the given interval? select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.\n\na. the absolute minimum/minima is/are $\\square$ at $x=\\square$\n\n(use a comma to separate answers as needed type an exact answer, using $\\pi$ as needed.)\n\nb. there is no absolute minimum of f on the given interval

determine the location and value of the absolute extreme values of f on the given interval, if they exist.\n\n$f(x)=\\cos 3x$ on $\\left-\\frac{\\pi}{4},\\frac{\\pi}{2}\\right$\n\nwhat is/are the absolute maximum/maxima of f on the given interval? select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.\n\na. the absolute maximum/maxima is/are 1 at $x = 0$\n\n(use a comma to separate answers as needed. type an exact answer, using $\\pi$ as needed)\n\nb. there is no absolute maximum of f on the given interval\n\nwhat is/are the absolute minimum/minima of f on the given interval? select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.\n\na. the absolute minimum/minima is/are $\\square$ at $x=\\square$\n\n(use a comma to separate answers as needed type an exact answer, using $\\pi$ as needed.)\n\nb. there is no absolute minimum of f on the given interval

Answer

Explanation:

Step1: Find the derivative of the function

The derivative of (y = \cos(3x)) using the chain rule ((\cos(u))^\prime=-\sin(u)\cdot u^\prime) (where (u = 3x) and (u^\prime=3)) is (y^\prime=-3\sin(3x)).

Step2: Find the critical points

Set (y^\prime = 0), so (-3\sin(3x)=0). Then (\sin(3x)=0), which gives (3x = k\pi), (x=\frac{k\pi}{3}), (k\in\mathbb{Z}). For the interval (\left[-\frac{\pi}{4},\frac{\pi}{2}\right]), when (k = 0), (x = 0); when (k=- 1), (x=-\frac{\pi}{3}\notin\left[-\frac{\pi}{4},\frac{\pi}{2}\right]); when (k = 1), (x=\frac{\pi}{3}\in\left[-\frac{\pi}{4},\frac{\pi}{2}\right]).

Step3: Evaluate the function at critical points and endpoints

  • Evaluate (f(x)=\cos(3x)) at (x =-\frac{\pi}{4}): (f\left(-\frac{\pi}{4}\right)=\cos\left(-\frac{3\pi}{4}\right)=-\frac{\sqrt{2}}{2}).
  • Evaluate at (x = 0): (f(0)=\cos(0) = 1).
  • Evaluate at (x=\frac{\pi}{3}): (f\left(\frac{\pi}{3}\right)=\cos(\pi)=-1).
  • Evaluate at (x=\frac{\pi}{2}): (f\left(\frac{\pi}{2}\right)=\cos\left(\frac{3\pi}{2}\right)=0).

Answer:

For the absolute minimum: The absolute minimum/minima is/are (-1) at (x=\frac{\pi}{3})