Is $f (x) = \frac{1}{x} - \frac{1}{x^{3}} + \frac{1}{x^{5}}$ $f(x)=1/x-1/x^3+1/x^5$ increasing or decreasing at $x = 2$ $x=2$ ?

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Answer 1 · 2016-01-10T07:56:13

Decreasing.

$f'(x) = -1/x^2 + 3/x^4 - 5/x^6$

$f'(2) = -9/64$

Decreasing at $x = 2$ .

To see why, consider a small change in $x$ , $deltax$ , near the neighborhood of $x=2$ .

We can approximate $f(2+deltax)$ as

$f(2) + f'(2)deltax$ .

This approximation is most accurate for small values of $deltax$ .

Since $f'(2)$ is negative,

for sufficiently small values of $deltax > 0$ , $f(2+deltax)<f(2)$ ,

and for sufficiently small values of $deltax < 0$ , $f(2+deltax)>f(2)$ .

Therefore, $f(x)$ is decreasing at $x=2$ .

Is f(x)=1/x-1/x^3+1/x^5f(x)=1x−1x3+1x5f(x)=1/x-1/x^3+1/x^5 increasing or decreasing at x=2x=2x=2?