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 x2−y2=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=x21−y21<0
Now , x21−y21−16<−16<0 ⇒ Point P lies outside the hyperbola.
Hence, 2 tangents can be drawn.