Find-the-minimum-value-of-5-x-2-x-2-x-1-x-x-1-0-

Question Number 116348 by Dwaipayan Shikari last updated on 03/Oct/20

Find the minimum value of \frac{(5 + x) {(2 + x)}^{2}}{x (1 + x)} (x \neq - 1, 0)

Commented by MJS_new last updated on 03/Oct/20

- \infty < \frac{(5 + x) {(2 + x)}^{2}}{x (1 + x)} < \infty

\Rightarrow minimum is - \infty

or do you want local minima ?

Commented by Dwaipayan Shikari last updated on 03/Oct/20

Yes

Answered by MJS_new last updated on 03/Oct/20

y = \frac{{(x + 2)}^{2} (x + 5)}{x (x + 1)}

y^{'} = \frac{(x + 2) (x^{3} - 15 x - 10)}{x^{2} {(x + 1)}^{2}}

y^{″} = \frac{8 (4 x^{3} + 15 x^{2} + 15 x + 5)}{x^{3} {(x + 1)}^{3}}

y^{'} = 0

\Rightarrow

x_{1} \approx - 3.48261 \land y \approx . 385778; y^{″} < 0 \Rightarrow local \max

x_{2} = - 2 \land y = 0; y^{″} > 0 \Rightarrow local \min

x_{3} \approx - . 688417 \land y \approx - 34.5782; y^{″} < 0 \Rightarrow local \max

x_{4} \approx 4.17103 \land y \approx 16.1925; y^{″} > 0 \Rightarrow local \min

Commented by Dwaipayan Shikari last updated on 03/Oct/20

Thanking you

Commented by MJS_new last updated on 03/Oct/20

{you}^{'} re welcome

the exact values of the roots if the 3^{rd} degree

polynomial are not useful in this case

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