given that f(x)= x^3 − 3x^2 + ax + b and (x−1) is a factor of f(x) also the maximum value of f(x) at poin where x = 1 is 12 find a) (dy/d) (f(x)) b) the values of a and b c) factorise f(x) completely d) hence evaluate ∫


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Question Number 39700 by Rio Mike last updated on 10/Jul/18

$g i v e n t h a t f (x) = x^{3} - 3 x^{2} + a x + b$ $a n d (x - 1) i s a f a c t o r o f f (x)$ $a l s o t h e m a x i m u m v a l u e o f$ $f (x) a t p o i n w h e r e x = 1$ $i s 12 f i n d$ $a) \frac{d y}{d} (f (x))$ $b) t h e v a l u e s o f a a n d b$ $c) f a c t o r i s e f (x) c o m p l e t e l y$ $d) h e n c e e v a l u a t e \int_{3}^{4} [f (x)] d x$
Commented by jorge160895@hotmail.com last updated on 10/Jul/18

$I f (x - 1) i s a f a c t o r o f f (x) t h e n$ $f (1) = 0 = 1 - 3 + a + b \to a + b = 2$ $f^{'} (x) = 3 x^{2} - 6 x + a \land f^{'} (1) = 12$ $3 - 6 + a = 1 \to b) a = 4 \land b = - 2$ $a) f^{'} (x) = 3 x^{2} - 6 x + 4$ $c) x^{3} - 3 x^{2} + 4 x - 2 = (x - 1) (x^{2} - 2 x + 2)$ $d) \underset{3}{\int^{4}} f (x) = \frac{x^{4}}{4} - x^{3} + 2 x^{2} - 2 x + c \underset{3}{\overset{4}{∣}}$ $(24 + c) - (\frac{81}{4} - 15 + c) = \frac{75}{4}$
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