Question

find the value or values of c that satisfy the equation \\( \frac { f ( b ) - f ( a ) } { b - a } = f ^ { prime } ( c ) \\) in the conclusion of the mean value theorem for the given function and interval. \\( f ( x ) = \sqrt { x - 8 }, 8,13 \\) simplify your answer. use a comma to separate answers as needed.)

Explanation:

Step1: Calculate \( f(b) \) and \( f(a) \)

Given \( f(x)=\sqrt{x - 8} \), \( a = 8 \), \( b = 13 \).
\( f(a)=f(8)=\sqrt{8 - 8}=0 \)
\( f(b)=f(13)=\sqrt{13 - 8}=\sqrt{5} \)
Then \( \frac{f(b)-f(a)}{b - a}=\frac{\sqrt{5}-0}{13 - 8}=\frac{\sqrt{5}}{5} \)

Step2: Find \( f^{\prime}(x) \)

Using the power rule \( (x^n)^\prime=nx^{n - 1} \), for \( f(x)=(x - 8)^{\frac{1}{2}} \), \( f^{\prime}(x)=\frac{1}{2}(x - 8)^{-\frac{1}{2}}=\frac{1}{2\sqrt{x - 8}} \)

Step3: Solve for \( c \)

Set \( f^{\prime}(c)=\frac{f(b)-f(a)}{b - a} \), so \( \frac{1}{2\sqrt{c - 8}}=\frac{\sqrt{5}}{5} \)
Cross - multiply: \( 5 = 2\sqrt{5}\sqrt{c - 8} \)
Divide both sides by \( 2\sqrt{5} \): \( \frac{5}{2\sqrt{5}}=\sqrt{c - 8} \)
Simplify \( \frac{5}{2\sqrt{5}}=\frac{\sqrt{5}}{2} \) (rationalizing the denominator: \( \frac{5}{2\sqrt{5}}=\frac{5\sqrt{5}}{2\times5}=\frac{\sqrt{5}}{2} \))
Square both sides: \( (\frac{\sqrt{5}}{2})^2=c - 8 \)
\( \frac{5}{4}=c - 8 \)
Add 8 to both sides: \( c=8+\frac{5}{4}=\frac{32 + 5}{4}=\frac{37}{4}=9.25 \)

Answer:

\( c = \frac{37}{4} \)