Choose the correct answers from the given four options: For the f… - Vidyadip

MCQ

Choose the correct answers from the given four options:
For the function $\text{f(x)}=\text{x}+\frac{1}{\text{x}},\text{x}\in[1,3],$ the value of c for mean value theorem is:

A
1
✓
$\sqrt{3}$
C
2
D
None of these

✓

Answer

Correct option: B.

$\sqrt{3}$

$\text{f(x)}=\text{x}+\frac{1}{\text{x}},$ which is continuous and differentiable.
So, by mean value theorem there exists atleast one $\text{c}\in(1,3)$ such that

$\because\ \text{f}'\text{c}=\frac{\text{f(b)}-\text{f(a)}}{\text{b}-\text{a}}$

$\Rightarrow\ 1-\frac{1}{\text{c}^2}=\frac{\frac{10}{3}-2}{3-1}$

$\Rightarrow\ \frac{\text{c}^2-1}{\text{c}^2}=\frac{2}{3}$

$\Rightarrow\ 3(\text{c}^2-1)=2\text{c}^2$

$\Rightarrow\ 3\text{c}^2-2\text{c}^2=3$

$\Rightarrow\ \text{c}^2=3$

$\Rightarrow\ \text{c}=\sqrt{3}\in(1,3)$

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