find the extreme values of the function on the given interval.\n$f(x)=4\\sin(x)$ on $\\frac{\\pi}{6},\\frac{2…

find the extreme values of the function on the given interval.\n$f(x)=4\\sin(x)$ on $\\frac{\\pi}{6},\\frac{2\\pi}{3}$\nenter the maximimum value here, or enter none if there isnt one.\n\nenter the minimimum value here, or enter none if there isnt one.

find the extreme values of the function on the given interval.\n$f(x)=4\\sin(x)$ on $\\frac{\\pi}{6},\\frac{2\\pi}{3}$\nenter the maximimum value here, or enter none if there isnt one.\n\nenter the minimimum value here, or enter none if there isnt one.

Answer

Explanation:

Step1: Find the derivative of the function

The derivative of (f(x) = 4\sin(x)) is (f^\prime(x)=4\cos(x)).

Step2: Find the critical points

Set (f^\prime(x) = 0), so (4\cos(x)=0), which gives (\cos(x) = 0). On the interval (\left[\frac{\pi}{6},\frac{2\pi}{3}\right]), (\cos(x)=0) when (x=\frac{\pi}{2}).

Step3: Evaluate the function at critical points and endpoints

  • Evaluate (f(x)) at (x = \frac{\pi}{6}): (f\left(\frac{\pi}{6}\right)=4\sin\left(\frac{\pi}{6}\right)=4\times\frac{1}{2}=2).
  • Evaluate (f(x)) at (x=\frac{\pi}{2}): (f\left(\frac{\pi}{2}\right)=4\sin\left(\frac{\pi}{2}\right)=4\times1 = 4).
  • Evaluate (f(x)) at (x=\frac{2\pi}{3}): (f\left(\frac{2\pi}{3}\right)=4\sin\left(\frac{2\pi}{3}\right)=4\times\frac{\sqrt{3}}{2}=2\sqrt{3}\approx3.46).

Answer:

Maximum value: (4) Minimum value: (2)