Question 1208061
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Part (a)


You can use the method tutor greenestamps mentioned. <ul><li>If all exponents of a polynomial function are even, then the function is even.</li><li>If all exponents of a polynomial function are odd, then the function is odd.</li><li>If the polynomial function has a mix of even and odd exponents, then the function is neither even nor odd.</li></ul>Your polynomial function is
g(x) = -x^4 + 32x + 144
which is the same as writing
g(x) = -x^4 + 32x^1 + 144x^0
The exponents are 4, 1, and 0. We have a mix of even and odd exponents, so <font color=red>the function is neither even nor odd</font>


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Another approach for part (a)


Recall that:<ul><li>If g(x) = g(-x) for all x in the domain, then g(x) is even.</li><li>If g(-x) = -g(x) for all x in the domain, then g(x) is odd.</li></ul>The key thing to focus on here is the "for all x".
If we can find one counterexample, then we'll disprove the claim.


Use a calculator, or do so by hand, to determine that
g(1) = 175
g(-1) = 111


g(1) = g(-1) is false so g(x) = g(-x) isn't true for all x in the domain.
This means g(x) isn't an even function.


g(-1) = -g(1) being false means g(-x) = -g(x) is also false.
This means g(x) isn't odd.


We conclude that <font color=red>g(x) is neither even nor odd</font>



If you wanted you can determine these facts
g(x) = -x^4 + 32x + 144
g(-x) = -(-x)^4 + 32(-x) + 144
g(-x) = -x^4 - 32x + 144
and,
g(x) = -x^4 + 32x + 144
-g(x) = -(-x^4 + 32x + 144)
-g(x) = x^4 - 32x - 144
Comparing g(x) with g(-x) shows the two functions aren't identical (since the +32x turned into -32x). Therefore, g(x) isn't even.
Comparing g(-x) with -g(x) shows the two functions aren't identical. Therefore, g(x) isn't odd.
A graphing calculator can be very useful here.


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Part (b)


There are many graphing options.
Usually I would turn to <a href="https://www.geogebra.org/calculator">GeoGebra</a>, but <a href="https://www.desmos.com/calculator">Desmos</a> is actually the quicker choice when finding the local max.


Type the function into Desmos.
Adjust the window so you can see the highest point. 
The window I'm using is
xMin = -10
xMax = 10
yMin = -300
yMax = 300
Other window parameters are possible.
{{{graph(400,400,-10,11,-300,329,0,-x^4 + 32x + 144)}}}
The curve is a weird lumpy hill of sorts. 
The curve isn't symmetric, but it somewhat resembles a parabola.


To find the highest point simply click anywhere on the curve. 
A small dot at the peak of this curve should show up. 
Click on that highest point to have the coordinates show up. 
You may have to click more than once.


<font color=red>The highest point on this curve is at (2,192)</font>
Therefore, g(x) reaches its max value of 192 when x = 2.


I'm not sure why your teacher mentioned "There is a local maximum value of 400 at x = 4" when that is false.
It's possible your teacher made a typo somewhere?
Note how g(4) = 16 and you can find larger outputs when trying something like g(0) and g(1).
Also, this curve has one local max only. 
There aren't any neighborhoods we can zoom in on to get another local max.
The region -infinity < x < 2 is always increasing while 2 < x < infinity is always decreasing.


If you want to use GeoGebra, then you'll need to use the function called "max". The template is
max(f, a,b)
where f is the function and a,b are the endpoints of the interval you are searching.
One possible thing we could type in would be
max( -x^4 + 32x + 144, 0, 4)
The result GeoGebra produces is the point (1.9999999981, 192)
We get a bit of rounding error. The x coordinate should be exactly 2. 


If you want to use something like a TI83, then check out <a href="https://www.montgomerycollege.edu/_documents/academics/support/learning-centers/ackerman-learning-center-rockville/one-page-max-min.pdf">this article</a>
Caution: The "zoomFit" feature might be tempting to use, but it's better to type in the xMin,xMax,yMin,yMax values by hand. 


Want to do this without a graphing calculator? It would take a while, but you can make a table of values. 
It can be done entirely by hand (slowest method) or using a non-graphing calculator (slightly faster but still pretty slow).
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