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Question

If f:RR and g:RR are two functions such that f(x)+f''(x)=xg(x)f'(x) and g(x)>0 xR, 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)<0 x>0h(x)>0 x<0
So, x=0 is a point of maxima for h(x).

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