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Mean Value Theorems — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsMean Value Theorems4 Marks
Question
Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem.
f(x) = x2 - 2x + 4 on [1, 5]
✓

Answer

We have,

f(x) = x2 - 2x + 4

Since a polynomial function is everywhere continuous and differentiable.

Therefore, f(x) is continuous on [1, 5] and differentiable on (1, 5).

Thus, both conditions of Lagrange's mean value theorem are satisfied.

So, there must exist at least one real number $\text{c}\in(1,5)$ such that

$\text{f}'(\text{c})=\frac{\text{f}(5)-\text{f}(-1)}{5-1}=\frac{\text{f}(5)-\text{f}(-1)}{4}$

Now, f(x) = x2 - 2x + 4

⇒ f'(x) = 2x - 2

⇒ f(5) = 25 - 10 + 4 = 19

⇒ f(1) = 1 - 2 + 4 = 3

$\therefore\ \text{f}'(\text{x})=\frac{\text{f}(5)-\text{f}(-1)}{4}$

$\Rightarrow2\text{x}-2=\frac{19-3}{4}$

$\Rightarrow2\text{x}-2-4=0$

$\Rightarrow\text{x}=\frac{6}{2}=3$

Thus, $\text{c}=3\in(1,5)$ such that $\text{f}'(\text{c})=\frac{\text{f}(5)-\text{f}(-1)}{5-1}$

Hence, Lagrange's mean value theorem is verified.

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