Let f be differentiable for all x.

Question

Let $f$ be differentiable for all $x$. If $f(1)=-2$ and $f'(x)\ge2$ for $x\in[1,6]$, then

Accepted Answer

As $$\,\,f\left( 1 \right) =  - 2\,\,\& \;\,f'\left( x \right) \ge 2\,\forall x \in \left[ {1,6} \right]$$

Applying Lagrange's mean value theorem

$${{f\left( 6 \right) - f\left( 1 \right)} \over 5} = f'\left( c \right) \ge 2$$

$$ \Rightarrow f\left( 6 \right) \ge 10 + f\left( 1 \right)$$

$$ \Rightarrow f\left( 6 \right) \ge 10 - 2$$

$$ \Rightarrow f\left( 6 \right) \ge 8.$$

Solutions · 1