Q59 / 150 · MHT CET 2023 (11 May Shift 1) · Mathematics

Value of $c$ satisfying the conditions and conclusions of Rolle's theorem for the function $\mathrm{f}(x)=x \sqrt{x+6}, x \in[-6,0]$ is

  • Correct answer
    A
    $-4$
  • B
    $4$
  • C
    $3$
  • D
    $-3$

Solution

$$ \begin{aligned} \mathrm{f}(x) & =x \sqrt{x+6} \\ \therefore \quad \mathrm{f}^{\prime}(x) & =x\left(\frac{1}{2 \sqrt{x+6}}\right)+\sqrt{x+6}(1) \\ & =\frac{x}{2 \sqrt{x+6}}+\sqrt{x+6} \end{aligned} $$ Since $\mathrm{f}(x)$ satisfies all the conditions of Rolle's Theorem, There exists $c \in(-6,0)$ such that $$ \begin{aligned} & \mathrm{f}^{\prime}(\mathrm{c})=0 \\ & \Rightarrow \frac{\mathrm{c}}{2 \sqrt{\mathrm{c}+6}}+\sqrt{\mathrm{c}+6}=0 \\ & \Rightarrow \mathrm{c}+2 \mathrm{c}+12=0 \\ & \Rightarrow \mathrm{c}=-4 \end{aligned} $$

Hence, the correct answer is option: 'A'.

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