If f:R→R and g:R→R are two functions such that f(x)+f''(x)=−xg(x)f'(x) and g(x)>0∀x∈R, then the function f2(x)+(f'(x))2 has
A
a maxima at x=0
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B
a minima at x=0
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C
a point of inflexion at x=0
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D
None of these
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Solution
The correct option is A a maxima at x=0 f(x)+f′′(x)=−xg(x)f′(x)
Let h(x)=f2(x)+(f′(x))2 ∴h′(x)=2f(x)f′(x)+2f′(x)f′′(x) =2f′(x)(−x)g(x)f′(x) =−2x(f′(x))2g(x)
As, h′(x)<0x>0h′(x)>0x<0
So, x=0 is a point of maxima for h(x).