Q43 / 90 · JEE Main 2012 (26 May Online) · Mathematics

Let $f:(-\infty, \infty) \rightarrow(-\infty, \infty)$ be defined by $f(x)=x^3+1$ Statement 1: The function fhas a local extremum at $x=0$ Statement 2: The function $f$ is continuous and differentiable on $(-\infty, \infty)$ and $f^{\prime}(0)=0$

  • A
    Statement 1 is true, Statement 2 is false.
  • B
    Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.
  • C
    Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation for Statement 1.
  • Correct answer
    D
    Statement 1 is false, Statement 2 is true.

Solution

Let $f:(-\infty, \infty) \rightarrow(-\infty, \infty)$ be defined by $f(x)=x^3+1$. Clearly, $f(x)$ is symmetric along $y=1$ and it has neither maxima nor minima. $\therefore$ Statement - 1 is false.

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

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