Izabelle Lowery

2022-10-20

Question about finding where the function increases and decreases on $f\left(x\right)=\frac{1}{x}$.
$f\left(x\right)=\frac{1}{x},x\ge 1$
I have been staring at this equation for a bit. Things I'm confused on.
the derivative of this is: ${f}^{\prime }\left(x\right)=\frac{-1}{{x}^{2}}$ now, how am I supposed to find where this derivative increases/decreases? Do I find the critical points first? by setting the derivative to 0? or do I solve it like $\frac{-1}{{x}^{2}}>0$ cross multiply to make it: $-1>{x}^{2}$ and if so once I square this does it make the result $x=-1,x=1$? I'm really lost here and it seems like it should be easier.
Does setting the derivative to > or < or = and solving for the x give a critical point?

Rene Jordan

Expert

Explanation:
Recall that intervals of increase and decrease of f correspond to intervals of positivity and negativity of f′, and critical points of f are where the roots of f′ are. Try graphing the function $-\frac{1}{{x}^{2}}$ using software. Where is the function positive, where is it negative, and where are its roots (if it has any)?

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