Ques3) If a differentiable function f (x) has arelative minimum at x = 0, then the function y = f(x) = ax²+bx+c has a relative minimum at x=0, then which of the following is / are correct :-

(a) b = 0 , all a (b) b=0, a> -(1/2) f ' (0) (c) a=0, b=0 (d) a=0 , b=0, c=2.

Y = ax² + bx + c

has max /min at f '(x) = 0

2ax + b = 0

at x=0, we have min. so b=0

also it has rel min. at x = 0

so f "(x) > 0

so ,2a > 0 i.e., a > 0

f '(0) = b =0

so a > -(1/2) f '(0)

a>O

option B

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Naga Ramesh
IIT Kgp - 2005 batch

f (x) has  arelative minimum at x = 0

That means

f '(0) = 0

or

b=0

.'. f (x) = ax²+c

Also since it is a minima

.'. f"(0) > 0

.'. a>0

Hence the answer is (b)