NDA 1 2024 Maths, question 92

Topic: Differential calculus

Passage · Q91 to Q92

A differentiable function $f(x)$ has a local maximum at $x=0$ . Let $y=2f(x)+ax-b$ .

Q92

The function $y$ has a relative maxima at $x=0$ for

(a) $a>0,b=0$
(b) for all $b$ and $a=0$ Correct answer
(c) for all $b>0$ only
(d) for all $a$ and $b=0$

Solution.

Answer: (b) for all $b$ and $a=0$

At 0, y′(0) = 2f′(0) + a = a, so a local maximum requires a = 0.
With a = 0, multiplying f by positive 2 preserves its local maximum, and subtracting any constant b does not change its location.
Thus every b is allowed when a = 0.