JEE Main 2020 (Online) 4th September Morning ShiftQuestion 1
Let $f$ be a twice differentiable function on $(1,6)$. If $f(2)=8$, $f'(2)=5$, $f'(x)\ge1$ and $f''(x)\ge4$, for all $x\in(1,6)$, then :
Differentiation and its Application
Solution
Concept. $f''\ge4$ gives a lower bound for $f'$, and integrating that gives a lower bound for $f$. Formulas used. (1) $f'(x)=f'(2)+\int_2^xf''\ge5+4(x-2)=4x-3$ for $x>2$(2) $f(5)=f(2)+\int_2^5f'(x)\,dx$Step 1 - $f'(5)$. By (1), $f'(5)\ge17$. Step 2 - $f(5)$. By (2) and (1), $$f(5)\ge8+\int_2^5(4x-3)\,dx=8+\left[2x^2-3x\right]_2^5=8+33=41.$$ Step 3 - Options. (B): $f(5)+f'(5)\ge58\ge28$. True. (A), (D) fail since $f(5)\ge41$. (C) fails since $f'(5)+f''(5)\ge17+4=21$. Answer: (B)