Question

by inspecting the graph of the function, find the absolute maximum and absolute minimum on the given interval.
( p ( x ) = - sqrt 3 { x } + 1 ) on (27,64)
choose the correct graph below.
find the absolute minimum. select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
a. the absolute minimum is at ( x = )
(use a comma to separate answers as needed )
b. there is no absolute minimum.

Explanation:

Step1: Analyze the function's monotonicity

The function \( p(x)=-\sqrt[3]{x}+1\) can be rewritten as \( p(x)=-x^{\frac{1}{3}} + 1\).
The derivative \( p^{\prime}(x)=-\frac{1}{3}x^{-\frac{2}{3}}=-\frac{1}{3x^{\frac{2}{3}}}\).
Since \( x\in[27,64]\), \( p^{\prime}(x)<0\) for all \( x\in(27,64)\). So the function \( p(x)\) is decreasing on the interval \([27,64]\).

Step2: Evaluate the function at the endpoints

When \( x = 27\), \( p(27)=-\sqrt[3]{27}+1=-3 + 1=-2\).
When \( x = 64\), \( p(64)=-\sqrt[3]{64}+1=-4 + 1=-3\).

Answer:

The absolute minimum is \(-3\) at \(x = 64\).