A function is matched below against an interval where it is supposed to be increasing. Which of the…

A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched? $Interval \rightarrow Function$
  1. $(-\infty, \infty) \rightarrow x^3-3 x^2+3 x+3$
  2. $[2, \infty) \rightarrow 2x^3-3 x^2-12 x+6$
  3. $\left(-\infty, \frac{1}{3}\right] \rightarrow \quad 3 x^2-2 x+1$
  4. $(-\infty, -4] \rightarrow x^3+6 x^2+6$

Solution

Clearly function $f(x)=3 x^2-2 x+1$ is increasing when $f^{\prime}(x)=6 x-2 \geq 0 \Rightarrow x \in[1 / 3, \infty)$ Hence $(C)$ is incorrect.

Asked in: JEE Main 2005

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