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Calculus
Level
2
A non zero polynomial with real coefficients has the property that f(x) = f'(x).f''(x) . Then , the leading coefficient of f(x) is
1/6
1/18
1
1/12
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1 solution
Note that: When we differentiate a polynomial, its degree decreases by 1.
Suppose degree of f ( x ) be n.
Since: f ( x ) = f ′ ( x ) . f ′ ′ ( x )
Therefore highest power (LHS) = highest power (RHS)
We get: n = ( n − 1 ) + ( n − 2 ) => n = 3
Now we assume a cubic: f ( x ) = a x 3 + b x 2 + c x + d
Substituting in f ( x ) = f ′ ( x ) . f ′ ′ ( x )
We have a x 3 + b x 2 + c x + d = ( 3 a x 2 + 2 b x + c ) . ( 6 a x + 2 b )
Equating coefficients of x 3 both sides. a = 1 8 a 2
a can't be 0. Therefore a = ( 1 / 1 8 )