intervals using 1st derivative test??

**monolith** · August 13th 2008, 08:50 PM

Find the intervals of inc and dec & local max and min of h(x)=x^3/2-256x? using 1st derivative test.?

thanks...this one is really getting me!!!

**Chris L T521** · August 13th 2008, 08:53 PM

Originally Posted by monolith

Find the intervals of inc and dec & local max and min of h(x)=x^3/2-256x? using 1st derivative test.?

thanks...this one is really getting me!!!

Is it $h(x)=x^{\frac{3}{2}}-256x$ or $h(x)=\tfrac{1}{2}x^3-256x$ ??

--Chris

**monolith** · August 13th 2008, 08:57 PM

i guess i wasnt using proper scripting...it is the first one you have written

$h(x)=x^{\frac{3}{2}}-256x$

**Chris L T521** · August 13th 2008, 09:49 PM

Originally Posted by monolith

i guess i wasnt using proper scripting...it is the first one you have written

$h(x)=x^{\frac{3}{2}}-256x$

First find $h'(x)$ :

$h'(x)=\tfrac{3}{2}x^{\frac{1}{2}}-256$

Now set this equal to zero:

$\tfrac{3}{2}x^{\frac{1}{2}}-256=0\implies x^{\frac{1}{2}}=\tfrac{512}{3}\implies x=\tfrac{262144}{9}\approx 29127.1$

This is the only critical point. So we investigate the behavior of $h'(x)$ over the intervals $\left(0,\tfrac{262144}{9}\right)\text{ and }\left(\tfrac{262144}{9},\infty\right)$

Pick a point in these intervals and plug them into $h'(x)$ , for simplicity, I'll pick 1 and 360,000 (the last one seems large, but when we take the square root of it, we get a nice number to work with...no decimals involved).

$h'(1)=\tfrac{3}{2}(1)^{\frac{1}{2}}-256=\color{red}\boxed{-254.5}$

It is decreasing here.

$h'(360,000)=\tfrac{3}{2}(360,000)^{\frac{1}{2}}-256=\color{red}\boxed{644}$

It is increasing here.

So to answer the first part of the question, the interval where its:

increasing -- $\left(\tfrac{262144}{9},\infty\right)$
decreasing -- $\left(0,\tfrac{262144}{9}\right)$

where $\tfrac{262144}{9}\approx 29127.111$

Now, take note that a maximum occurs when a function goes from increasing to decreasing at a critical point, and a minimum occurs when a function goes from decreasing to increasing at a critical point. Which one do you think it is? What is the coordinates of the maximum/minimum? I leave that for you to do.

I hope this makes sense!

--Chris

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