A calculus problem by Mohamad Zare

Calculus Level 3

f ( x ) = a x 3 + b x 2 + c x f(x)=ax^{3}+bx^{2}+c{x} if this function have three distinct real roots p , q , r p,q,r , then find f ′ ′ ( p + q + r ) f''(p+q+r) in terms of a,b or c.

6 c + 2 b 6c+2b − 4 b -4b 4 a c 4ac 6 a b 6ab − 6 b -6b b 2 − 4 a c b^{2}-4ac

Mohamad Zare by Mohamad Zare , 23, Iran

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1 solution

Mohamad Zare
Mar 25, 2016

f ′ ( x ) = 3 a x 2 + 2 b x f'(x)=3ax^{2}+2bx then f ′ ′ ( x ) = 6 a x + 2 b f''(x)=6ax+2b = 6 a × − b / a + 2 b 6a×-b/a+2b = − 4 b -4b , f ( x ) = x ( a x 2 + b x + c ) f(x)=x(ax^{2}+bx+c) so p + q + r p+q+r =0+ − b a \frac{-b}{a}

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