How do you use the first and second derivatives to sketch $f(x)=(x+2)/(x-3)$ ?

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How do you use the first and second derivatives to sketch $f(x)=(x+2)/(x-3)$ ?

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Answer 1 · 2017-01-08T18:19:37

graph{(x+2)/(x-3) [-15.58, 24.42, -8.16, 11.84]}

Explanation:

$f(x) = (x+2)/(x-3)$

$f'(x) = frac ( x-3 -x -2) ((x-3)^2) = -5/((x-3)^2)$

$f''(x) = 10/((x-3)^3)$

We can now analyse the behaviour of the derivatives to sketch the function:

(1) $f'(x) < 0$ everywhere in its domain $RR -{3}$ , so $f(x)$ is strictly decreasing and has no local extrema.

(2) for $x < 3$ , $f''(x) < 0$ so $f(x)$ is concave down in $(-oo,3)$

(3) for $x > 3$ , $f''(x) > 0$ so $f(x)$ is concave up in $(3,+oo)$

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How do you use the first and second derivatives to sketch $f(x)=(x+2)/(x-3)$ ?

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Humanities

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How do you use the first and second derivatives to sketch f(x)=(x+2)/(x-3) f(x)=(x+2)/(x-3)?

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Explanation:

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How do you use the first and second derivatives to sketch $f(x)=(x+2)/(x-3)$ ?