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$f(x)=2 x^{\\frac{1}{2}}-x$ on $0,4$\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 $\\square$ at $x=\\square$\n(use a comma to separate answers as needed.)\n\nb. there is no absolute maximum 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$f(x)=2 x^{\\frac{1}{2}}-x$ on $0,4$\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 $\\square$ at $x=\\square$\n(use a comma to separate answers as needed.)\n\nb. there is no absolute maximum of f on the given interval.

Answer

Explanation:

Step1: Find the derivative of (f(x))

Using the power rule, if (y = ax^{n}), then (y^\prime=anx^{n - 1}). For (f(x)=2x^{\frac{1}{2}}-x), (f^\prime(x)=2\times\frac{1}{2}x^{\frac{1}{2}-1}-1=x^{-\frac{1}{2}}-1=\frac{1}{\sqrt{x}}-1).

Step2: Find the critical points

Set (f^\prime(x) = 0), (\frac{1}{\sqrt{x}}-1 = 0). Add (1) to both sides: (\frac{1}{\sqrt{x}}=1). Cross - multiply: (\sqrt{x}=1), then (x = 1). Also, check the endpoints (x = 0) and (x = 4).

Step3: Evaluate (f(x)) at critical points and endpoints

  • When (x = 0): (f(0)=2\times0^{\frac{1}{2}}-0=0).
  • When (x = 1): (f(1)=2\times1^{\frac{1}{2}}-1=2 - 1=1).
  • When (x = 4): (f(4)=2\times4^{\frac{1}{2}}-4=2\times2 - 4=0).

Answer:

A. The absolute maximum/maxima is/are (1) at (x = 1)