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CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY4 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) = x3 - 2x2 - x + 3 on [0, 1]
✓

Answer

Here,

f(x) = x3 - 2x2 - x + 3

Since f(x) is polynomial function. So, f(x) is continuous in [0, 1] and differentiable in (0, 1).

Thus, both conditions of Lagrange's mean value theorem is appplicable.

Thus, there exist a point $\text{c}\in(0,1)$ such that

$\text{f}'(\text{c})=\frac{\text{f}(1)-\text{f}(0)}{1-0}$

$\Rightarrow3\text{c}^2-4\text{c}-1=\frac{[(1)^3-2(1)^2-(1)+3]-3}{1}$

⇒ 3c2 - 4c - 1 = 1-2

⇒ 3c2 - 4c + 1 = 0

⇒ 3c2 - 3c - c + 1 = 0

⇒ 3c(c - 1) - 1(c - 1) = 0

⇒ (3c - 1)(c - 1) = 0

$\Rightarrow\text{c}=\frac{1}{3}\in(0,1)$

Hence, Lagrange's mean value theorem is verified.

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