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Question

Let f′′(x)+f(x)+(f(x))2=x2 be the differential equation of a curve y=f(x) and let P be the point of local maximum of y=f(x). Then the number of tangents which can be drawn from P to x2y2=16 is

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Solution

At point P(x1,y1) of local maximum,
f(x1)=0 and f′′(x1)<0
f′′(x1)=x21(f(x1))2=x21y21<0
Now ,
x21y2116<16<0
Point P lies outside the hyperbola.
Hence, 2 tangents can be drawn.

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