calculate-If-f-x-x-2-1-x-2-x-2-x-4-1-x-2-2x-3-16-1-3-x-2-3-1-x-x-2-then-f-1-m-n-

Question Number 166829 by mnjuly1970 last updated on 01/Mar/22

c a l c u l a t e

I f, f (x) = \frac{(x^{2} + 1) \sqrt[3]{(x^{2} + x - 2) (x^{4} - 1) (x^{2} + 2 x - 3) + 16} + \sqrt{x^{2} + 3}}{(1 + x + x^{2})}

t h e n, f^{'} (1) = ? m . n

Answered by mathsmine last updated on 01/Mar/22

f (x) = \frac{(x^{2} + 1) \sqrt[3]{g (x)} + \sqrt{x^{2} + 3}}{(1 + x + x^{2})}

f^{'} (x) = \frac{(2 x \sqrt[3]{g (x)} + \frac{g^{'} (x)}{3 g^{\frac{2}{3}} (x)} (x^{2} + 1) + \frac{x}{\sqrt{x^{2} + 3}})}{1 + x + x^{2}} - \frac{(1 + 2 x) ((x^{2} + 1) \sqrt[3]{g (x)} + \sqrt{x^{2} + 3})}{{(1 + x + x^{2})}^{2}}

f^{'} (1) = \frac{2 \sqrt[3]{16} + 2 \frac{g^{'} (1)}{3 {(16)}^{\frac{2}{9}}} + \frac{1}{2}}{3} - \frac{2 \sqrt[3]{16} + 2}{3}

g (x) = (x^{2} + x - 2) (x^{4} - 1) (x^{2} + 2 x - 3) + 16

{(f g h)}^{^{'}} = f^{'} g h + {f g}^{'} h + {f g h}^{'}

1 r o o t o f x^{2} + x - 2, x^{4} - 1, x^{2} + 2 x - 3 \Rightarrow

g^{'} (1) = 0

f^{'} (1) = \frac{2 \sqrt[3]{16} + \frac{1}{2}}{3} - \frac{2 \sqrt[3]{16} + 2}{3} = - \frac{1}{2}