If 9 + f ′′ ( x ) + f ′ ( x ) = x 2 + f ( x ) 2 is the differential equation of a curve and P is a point of local minimum with f ′′ ( x ) 0 , how many tangents can be drawn from P to the circle x 2

Question

If 9 + f ′′ ( x ) + f ′ ( x ) = x 2 + f ( x ) 2 is the differential equation of a curve and P is a point of local minimum with f ′′ ( x ) > 0 , how many tangents can be drawn from P to the circle x 2 + y 2 = 9 ?

TestHub · Accepted Answer

At a local minimum f ′ ( x ) = 0 and with f ′′ ( x ) > 0 , we get x 2 + f ( x ) 2 = 9 + f ′′ ( x ) > 9 , so P lies strictly outside the circle. From an exterior point exactly two tangents can be drawn to the circle. Answer: 2.

Mathematics - Circle Question with Solution | TestHub

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