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Question Number 39416 by Rio Mike last updated on 06/Jul/18

G i v e n t h a t f (x) i s a c u b i c f u n c t i o n

a n d f (x) = x^{3} - \frac{x^{2}}{4} + 5 x - 7

a) f i n d o n e f a c t o r o f f (x)

b) f i n d \frac{d^{2} y}{{d x}^{2}} f o r f (x)

c) h e n c e E v a l u a t e y = \int_{0}^{\infty} f (x) .

Answered by MJS last updated on 06/Jul/18

(b) \frac{d^{2} y}{{d x}^{2}} [f (x)] = f^{″} (x) = 6 x - \frac{1}{2}

(c) \int f (x) d x = \frac{1}{4} x^{4} - \frac{1}{12} x^{3} + \frac{5}{2} x^{2} - 7 x + C \Rightarrow

\underset{0}{\int^{\infty}} f (x) d x = + \infty

(a) what are we allowed to use ?

approximation procedure with a calculator

gives x \approx 1.15710

Cardano :

x^{3} - \frac{1}{4} x^{2} + 5 x - 7 = 0

z = x + \frac{1}{12}

z^{3} + \frac{239}{48} z - \frac{5689}{864} = 0

u = \frac{1}{12} \sqrt[3]{5689 + 24 \sqrt{79890}}; v = \frac{1}{12} \sqrt[3]{5689 - 24 \sqrt{79890}}

z_{1} = u + v; x_{1} = \frac{1}{12} + u + v \approx 1.15710

z_{2} = (- \frac{1}{2} + \frac{\sqrt{3}}{2} i) u + (- \frac{1}{2} - \frac{\sqrt{3}}{2} i) v

z_{2} = (- \frac{1}{2} - \frac{\sqrt{3}}{2} i) u + (- \frac{1}{2} + \frac{\sqrt{3}}{2} i) v

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