find the absolute extrema of the function, if they exist, over the indicated interval. also indicate the x…

find the absolute extrema of the function, if they exist, over the indicated interval. also indicate the x - value at which each extremum occurs. if no interval is specified, use the real numbers. (-∞, ∞)\nn(x)=\\sqrt7{x}, 0,128\nselect the correct choice below and, if necessary, fill in the answer boxes to complete your choice\na. the absolute maximum is at x = and the absolute minimum is at x =\n(use a comma to separate answers as needed )\nb. the absolute minimum is at x = and there is no absolute maximum\n(use a comma to separate answers as needed )\nc. the absolute maximum is at x = and there is no absolute minimum\n(use a comma to separate answers as needed )\nd. there is no absolute maximum and no absolute minimum

find the absolute extrema of the function, if they exist, over the indicated interval. also indicate the x - value at which each extremum occurs. if no interval is specified, use the real numbers. (-∞, ∞)\nn(x)=\\sqrt7{x}, 0,128\nselect the correct choice below and, if necessary, fill in the answer boxes to complete your choice\na. the absolute maximum is at x = and the absolute minimum is at x =\n(use a comma to separate answers as needed )\nb. the absolute minimum is at x = and there is no absolute maximum\n(use a comma to separate answers as needed )\nc. the absolute maximum is at x = and there is no absolute minimum\n(use a comma to separate answers as needed )\nd. there is no absolute maximum and no absolute minimum

Answer

Explanation:

Step1: Find the derivative of the function

The function is (N(x)=\sqrt[7]{x}=x^{\frac{1}{7}}). Using the power rule (y = x^n), (y^\prime=nx^{n - 1}), we have (N^\prime(x)=\frac{1}{7}x^{\frac{1}{7}-1}=\frac{1}{7}x^{-\frac{6}{7}}=\frac{1}{7x^{\frac{6}{7}}}). The derivative (N^\prime(x)) is never zero (since the numerator is non - zero for all (x)) and is undefined at (x = 0). But (x = 0) is an endpoint of the interval ([0,128]).

Step2: Evaluate the function at the endpoints

Evaluate (N(x)) at (x = 0) and (x=128). When (x = 0), (N(0)=\sqrt[7]{0}=0). When (x = 128), (N(128)=\sqrt[7]{128}). Since (128 = 2^7), then (N(128)=\sqrt[7]{2^7}=2).

Answer:

A. The absolute maximum is (2) at (x = 128) and the absolute minimum is (0) at (x=0)