Verify mean value theorem for the function: f(x)= 25-x^2 in [1,5]… - Vidyadip

Question

Verify mean value theorem for the function:
$\text{f(x)}=\sqrt{25-\text{x}^2}\text{ in }[1,5].$

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Answer

We have, $\text{f(x)}=\sqrt{25-\text{x}^2}\text{ in }[1,5]$
Since $25 - x^2$ and square root function are continuous and differentiable in their domain, given function f(x) is also continuous and differentiable.
So, conditions of mean value thecorem are satisfied.
Hence, there exists atleast one $\text{c}\in(1,5)$ such that,
$\text{f}'(\text{c})=\frac{\text{f}(5)-\text{f}(1)}{5-1}$
$\Rightarrow\ \frac{-\text{c}}{\sqrt{25-\text{c}^2}}=\frac{0-\sqrt{24}}{4}$
$\Rightarrow\ 16\text{c}^2=24(25-\text{c}^2)$
$\Rightarrow\ 40\text{c}^2=600$
$\Rightarrow\ \text{c}^2=15$
$\Rightarrow\ \text{c}=\sqrt{15}\in(1,5)$
Hence, mean value theorem has been verified.

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