What are the local extrema, if any, of $f (x) = \frac{x^{3} + 3 x^{2}}{x^{2} - 5 x}$ $f (x) =(x^3+3x^2)/(x^2-5x)$ ?

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What are the local extrema, if any, of $f (x) = \frac{x^{3} + 3 x^{2}}{x^{2} - 5 x}$ $f (x) =(x^3+3x^2)/(x^2-5x)$ ?

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Answer 1 · 2018-06-03T12:48:13

$M I N (5 + 2 \sqrt{10}, 4 \sqrt{10} + 13)$ $MIN(5+2sqrt(10),4sqrt(10)+13)$ and $M A X (5 - 2 \cdot \sqrt{10}, - 4 \sqrt{10} + 13)$ $MAX(5-2*sqrt(10),-4sqrt(10)+13)$

Explanation:

By the Quotient rule we get

$f'(x)=((3x^2+6x)(x^2-5x)-(x^3+3x^2)(2x-5))/(x^2-5x)^2$
simplifying we obtain

$f'(x)=(x^2-10x-15)/(x-5)^2$

$f''(x)=80/(x-5)^3$

so we have

$f''(5+2*sqrt(10))=80/(2sqrt(10))^3>0$

$f''(5-2sqrt(10))=80/(-2sqrt(10))^3<0$

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What are the local extrema, if any, of $f (x) = \frac{x^{3} + 3 x^{2}}{x^{2} - 5 x}$ $f (x) =(x^3+3x^2)/(x^2-5x)$ ?

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Explanation:

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What are the local extrema, if any, of f (x) =(x^3+3x^2)/(x^2-5x)f(x)=x3+3x2x2−5xf (x) =(x^3+3x^2)/(x^2-5x)?

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What are the local extrema, if any, of $f (x) = \frac{x^{3} + 3 x^{2}}{x^{2} - 5 x}$ $f (x) =(x^3+3x^2)/(x^2-5x)$ ?