Q110 / 160 · TS EAMCET 2024 (09 May Shift 1) · Mathematics

If $f(x)=(2 x-1)(3 x+2)(4 x-3)$ is a real valued function defined on $\left[\frac{1}{2}, \frac{3}{4}\right]$, then the value(s) of ' $c$ ' as defined in the statement of Rolle's theorem

  • A
    Does not exist
  • B
    $\frac{7 \pm \sqrt{247}}{36}$
  • C
    $\frac{7-\sqrt{247}}{36}$
  • Correct answer
    D
    $\frac{7+\sqrt{247}}{35}$

Solution

$f(x)=(2 x-1)\left(12 x^2-x-6\right)$ $\begin{aligned} & =24 x^3-2 x^2-12 x-12 x^2+x+6 \\ & =24 x^3-14 x^2-11 x+6 \\ & f^{\prime}(x)=72 x^2-28 x-11 \\ & f^{\prime}(c)=72 c^2-28 c-11=0 \\ & \Rightarrow c=\frac{28 \pm \sqrt{28^2+11 \times 4 \times 72}}{144} \\ & =\frac{28 \pm 4 \sqrt{49+11 \times 18}}{144}=\frac{7 \pm \sqrt{247}}{36} \\ & \Rightarrow c=\frac{7+\sqrt{247}}{36} \in\left[\frac{1}{2}, \frac{3}{4}\right] .\end{aligned}$

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

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