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If f(x) is a quadratic expression such that f(x)>0 for every x belongs to R,and if g(x)=f(x)+f'(x)+f"(x),then prove that g(x)>0 for every x belonging to R.

If f(x) is a quadratic expression such that f(x)>0 for every x belongs to R,and if g(x)=f(x)+f'(x)+f"(x),then prove that g(x)>0 for every x belonging to R.

Grade:12th Pass

1 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
6 years ago
Ans:
f(x) = ax^{2}+bx+c
f(x) > 0
\Rightarrow \Delta < 0
\Rightarrow b^{2} < 4ac
g(x) = f(x)+f'(x)+f''(x)
g(x) = ax^{2}+(b+2a)x+(2a+b+c) = 0
\Delta = (b+2a)^{2}-4.2a.(2a+b+c)
\Delta = b^{2}-4ac-4a^{2}
b^{2}-4ac<0
\Rightarrow \Delta <0
\Rightarrow g(x) >0
Thanks & Regards
Jitender Singh
IIT Delhi
askIITians Faculty

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