AIEEE 2005 (Offline)Question 1
Let $f$ be differentiable for all $x$. If $f(1)=-2$ and $f'(x)\ge2$ for $x\in[1,6]$, then
Differentiation and its Application
Solution
Concept. The mean value theorem turns $f'\ge2$ into a bound on $f(6)$. Formulas used. (1) MVT: $f(6)-f(1)=5f'(c)$, $c\in(1,6)$Step 1 - Bound. By (1), $f(6)-f(1)\ge5\cdot2=10$. Step 2 - Value. $f(6)\ge-2+10=8$. Answer: (A)