Q170 / 270 · NDA 2016 (Phase 1) · Mathematics

Consider the function $\mathrm{f}(\mathrm{x})=|\mathrm{x}-1|+\mathrm{x}^2$, where $\mathrm{x} \in \mathrm{R}$. Which one of the following statements is correct?

  • A
    $f(x)$ has local minima at more than one point in $(-\infty, \infty)$
  • B
    $\mathrm{f}(\mathrm{x})$ has local maxima at more than one point in $(-\infty, \infty)$
  • Correct answer
    C
    $f(x)$ has local minimum at one point only in $(-\infty, \infty)$
  • D
    $f(x)$ has neither maxima nor minima in $(-\infty, \infty)$

Solution

$f(x)$ has local minimum at one point only in $(-\infty, \infty)$. $f^{\prime}(x)=\left\{\begin{array}{lll} 2 x-1 & ; & x \lt 1 \\ 2 x+1 & ; & x \geq 1 \end{array}\right.$ Clearly; for $(\mathrm{x} \gt 1) ; \mathrm{f}^{\prime}(\mathrm{x}) \gt 0 \geq \&$ for $(\mathrm{x} \lt 1)$ $x=\frac{1}{2}$ is the point of local minima

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

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