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Question

If a differentiable function f(x) has a relative minimum at x = 0, then the function y = f(x) + ax + b has a relative minimum at x = 0 for


A

all a > 0

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B

all b > 0

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C

all a and b

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D

all b, if a = 0

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Solution

The correct option is D

all b, if a = 0


Since, f(x) has a relative minimum at x = 0
f'(0)=0 and f'' (0) > 0
Now, y = f(x) + ax + b
dydx=f(x)+a [dydx]x=0=f(0)+a=a=0, if a=0Also, d2ydx2=f′′(x) [d2ydx2]x=0=f′′(0)>0
y has a relative minimum at x = 0, if a = 0 and for all b.


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