Question:

The value of $c$ in Rolle's theorem for the function $f(x) = x^3 - 3x$ in the interval $\left[0, \sqrt{3}\right]$ is

Updated On: Jul 7, 2022
  • $1$
  • $-1$
  • $\frac{3}{2}$
  • $\frac{1}{3}$
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The Correct Option is A

Solution and Explanation

Since Rolle's theorem satisfy in $\left[0, \sqrt{3}\right]$. $\therefore f '\left(c\right)=0$ $\Rightarrow 3c^{2}-3=0$ $\Rightarrow c^{2}-1$ $\Rightarrow c \pm1$ $\Rightarrow c=1 \left(\because c=-1 \notin\left[0, \sqrt{3}\right]\right)$
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Rate of Change of Quantities:

If some other quantity ‘y’ causes some change in a quantity of surely ‘x’, in view of the fact that an equation of the form y = f(x) gets consistently pleased, i.e, ‘y’ is a function of ‘x’ then the rate of change of ‘y’ related to ‘x’ is to be given by 

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Increasing and Decreasing Function:

Consider y = f(x) be a differentiable function (whose derivative exists at all points in the domain) in an interval x = (a,b).

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