If Rolle's theorem holds for function f(x) = 2{x^3} + b{x^2} + cx,,x, ∈ ,left[ { - 1,1} right]

English

Gujarati

Hindi

5. Continuity and Differentiation

normal

If Rolle's theorem holds for the function $f(x) = 2{x^3} + b{x^2} + cx,\,x\, \in \,\left[ { - 1,1} \right]$ at the point $x = \frac{1}{2}$ , then $(2b+c)$ is equal to

A

$1$

B

$-1$

C

$2$

D

$-3$

Solution

$b=\frac{1}{2}, c=-2$

$\therefore \quad(2 b+c)=1+(-2)=-1$

(apply conditions of Rolle's theorem)

Standard 12

Mathematics

Share

Similar Questions

If from mean value theorem, $f'({x_1}) = {{f(b) – f(a)} \over {b – a}}$, then

normal

If $f(x) = \cos x,0 \le x \le {\pi \over 2}$, then the real number $ ‘c’ $ of the mean value theorem is

easy

If $f:[-5,5] \rightarrow \mathrm{R}$ is a differentiable function and if $f^{\prime}(x)$ does not vanish anywhere, then prove that $f(-5) \neq f(5).$

medium

Examine if Rolle's Theorem is applicable to any of the following functions. Can you say some thing about the converse of Roller's Theorem from these examples?

$f(x)=[x]$ for $x \in[5,9]$

medium

Examine if Rolle's Theorem is applicable to any of the following functions. Can you say some thing about the converse of Roller's Theorem from these examples?

$f(x)=x^{2}-1$ for $x \in[1,2]$

medium