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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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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)
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