The value of <math> <mi>c</mi> </math> in the Lagrange's mean value theorem for the function <math> <mi>f</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>=</mo> <msup> <mrow> <mrow> <mi>x</mi> </mrow> </mrow> <mrow> <mrow> <mn>3</mn> </mrow> </mrow> </msup> <mo>-</mo> <mn>4</mn> <msup> <mrow> <mrow> <mi>x</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <mn>8</mn> <mi>x</mi> <mo>+</mo> <mn>11</mn> </math> , when <math> <mi>x</mi> <mo>∈</mo> <mo>[</mo> <mn>0,1</mn> <mo>]</mo> </math> is:

Using LMVT f ' ( c ) = f ( 1 ) - f ( 0 ) 1 - 0 <!- [endif]-><!- [endif]-> 3 c 2 - 8 c + 8 = 16 - 11 1 - 0 <!- [endif]-><!- [endif]-> 3 c 2 - 8 c + 3 = 0 <!- [endif]-><!- [endif]-> c = 4 - 7 3 ∈ ( 0,1 ) <!- [endif]-><!- [endif]->

The value of <math> <mi>c</mi> </math> in the Lagrange's mean value theorem for the function <math> <mi>f</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>=</mo> <msup> <mrow> <mrow> <mi>x</mi> </mrow> </mrow> <mrow> <mrow> <mn>3</mn> </mrow> </mrow> </msup> <mo>-</mo> <mn>4</mn> <msup> <mrow> <mrow> <mi>x</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <mn>8</mn> <mi>x</mi> <mo>+</mo> <mn>11</mn> </math> , when <math> <mi>x</mi> <mo>∈</mo> <mo>[</mo> <mn>0,1</mn> <mo>]</mo> </math> &nbsp;is: &lt;!-- [if !supportLineBreakNewLine]--&gt; &lt;!--[endif]--&gt;

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Maths Class 12th Maths Continuity and Differentiability

The value of $c$ in the Lagrange's mean value theorem for the function $f (x) = x^{3} - 4 x^{2} + 8 x + 11$ , when $x \in [0,1]$ is:

Option 1 -

$\frac{\sqrt[]{7} - 2}{3}$

Option 2 -

$\frac{4 - \sqrt[]{5}}{3}$

Option 3 -

$\frac{4 - \sqrt[]{7}}{3}$

Option 4 -

$\frac{2}{3}$

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Answered by

alok kumar singh | Contributor-Level 10

a month ago

Correct Option - 3

Detailed Solution:

Using LMVT

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

$3 c^{2} - 8 c + 8 = \frac{16 - 11}{1 - 0}$

$3 c^{2} - 8 c + 3 = 0$

$c = \frac{4 - \sqrt[]{7}}{3} \in (0,1)$

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The value of $c$ in the Lagrange's mean value theorem for the function $f (x) = x^{3} - 4 x^{2} + 8 x + 11$ , when $x \in [0,1]$ is: