f(x) = 3x^3 − 2x + k has factor (x − 1) find the value of k. with these value evaluate a) (d/(dx ))(f(x)_ ) b) ∫


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Question Number 39144 by Rio Mike last updated on 03/Jul/18

$f (x) = 3 x^{3} - 2 x + k h a s$ $f a c t o r (x - 1)$ $f i n d t h e v a l u e o f k .$ $w i t h t h e s e v a l u e$ $e v a l u a t e$ $a) \frac{d}{d x} (f {(x)}_{})$ $b) \int_{5}^{2} (f (x))$
	Answered by MJS last updated on 03/Jul/18

	$(a) \frac{d}{d x} [f (x)] = f^{'} (x) = 9 x^{2} - 2$ $(x - 1) (x - α) (x - β) - (x^{3} - \frac{2}{3} x + \frac{k}{3}) = 0$ $α + β + 1 = 0$ $α + α β + β + \frac{2}{3} = 0$ $α β + \frac{k}{3} = 0$ $α = - β - 1$ $β^{2} + β + \frac{1}{3} = 0$ $β^{2} + β - \frac{k}{3} = 0$ $\Rightarrow k = - 1$ $f (x) = 3 x^{3} - 2 x - 1$ $(b) \underset{5}{\int^{2}} (3 x^{3} - 2 x - 1) d x = {[\frac{3}{4} x^{4} - x^{2} - x]}_{5}^{2} =$ $= 6 - \frac{1755}{4} = - \frac{1731}{4}$
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