Q103 / 225 · JEE Main 2005 · Mathematics

Let $f$ be differentiable for all $x$. If $f(1)=-2$ and $f^{\prime}(x) \geq 2$ for $x \in[1,6]$, then

  • Correct answer
    A
    $f(6) \geq 8$
  • B
    $f(6) < 8$
  • C
    $f(6) < 5$
  • D
    $f(6)=5$

Solution

As $f(1)=-2 \quad \& f^{\prime}(x) \geq 2 \quad \forall x \in[1,6]$ Applying Lagrange's mean value theorem $$ \begin{aligned} & \frac{f(6)-f(1)}{5}=f^{\prime}(c) \geq 2 \\ & \Rightarrow f(6) \geq 10+f(1) \\ & \Rightarrow f(6) \geq 10-2 \\ & \Rightarrow f(6) \geq 8 . \end{aligned} $$

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

Relevant Questions