Q223 / 270 · NDA 2022 (Phase 2) · Mathematics

What is the minimum value of the function $f(x)=\log _{10}\left(x^2+2 x+11\right) ?$

  • A
    $0$
  • Correct answer
    B
    $1$
  • C
    $2$
  • D
    $10$

Solution

$\begin{aligned} & \text { Since, }, f(x)=\log _{10}\left(x^2+2 x+11\right) \\ & \therefore f^{\prime}(x)=\frac{2 x+2}{x^2+2 x+11} \log _{10}^e=0 \\ & \Rightarrow x=-1(\text { critical point }) \\ & f^{\prime \prime}(x)=\frac{2\left(x^2+2 x+11\right)-(2 x+2)^2}{\left(x^2+2 x+11\right)^2} \cdot \log _{10}^e \\ & \because f^{\prime \prime}(-1) \gt 0 \\ & \therefore \quad \text { Minimum value of } f(x) \\ & f(-1)=\log _{10} 10=1\end{aligned}$

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

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