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If we distribute X to the one fifth out, we get X to the six
fifths plus X to the one fifth.
We take our first derivative.
We get 6 fifths, X to the 1/5 + 1/5, X to the -4 fifths.
Getting that to be one fraction, we've got to multiply the first
term by X to the 4/5 / X to the 4/5.
So we're going to get 6X plus one all over, 5X to the 4/5.
Figuring out where the first derivative is zero six, X + 1 =
0, X is -1 six.
Figuring out where the derivative is undefined, X to
the 4th, 5th equals 0X is 0.
Putting that on a number line going to the right of 02/3/10 a
hundred, we get a positive value if we do something in between
zero and negative 1/6 because that X to the 4/5 that four
tells us whatever we put in on either side would come out the
same sign.
So in this case, on the right side it's positive and on the
left side it's positive.
And then at the -1 sixth, it's going to do the opposite.
It's going to turn negative.
So the original function is decreasing from negative
Infinity to 0 or negative Infinity to -1 sixth.
It's increasing from -1 six to zero and also from zero to
Infinity.
So we could say -1 six to Infinity, it's increasing.
So there is no local Max.
The local min occurs at negative 1/6, and to get the Y value we
just stick negative 1/6 into the original equation and we get
approximately -.582.
So the function is increasing from negative 1/6 to Infinity,
and it's decreasing from negative Infinity to negative
1/6.