Question 1208061: Given G(x) = -x^4 + 32x + 144,
A. Determine if G is even, odd,or even.
I think I have to evaluate G(-x). Yes?
B.There is a local maximum value of 400 at x = 4.
Determine a second maximum value using a graph.
Note: DO NOT USE CALCULUS.
Found 3 solutions by greenestamps, math_tutor2020, ikleyn:
Answer by greenestamps(13367)   (Show Source):
You can put this solution on YOUR website!
It's generally worth your time and effort to check that you have typed your question correctly before you submit it.
"A. Determine if G is even, odd,or even."
To make any sense, the choices should be even, odd, or neither....
To answer your first question, no, you do NOT need to evaluate G(-x). You only need to look at the powers of the variables. The polynomial function is even if and only if all the powers are even; it is odd if and only if all the powers are odd.
The function you show has both even and odd powers, so it is neither even nor odd.
"B.There is a local maximum value of 400 at x = 4. Determine a second maximum value using a graph."
For the function you show, there is NOT a local maximum of 400 at x=4.
Furthermore, you can't find another local maximum by looking at the graph of the function you show. (In fact, if you graph your function, you can easily see that there is only one local maximum.)
The only way the instruction in part B makes sense is if the given function is even.
And, in fact, if the second term is 32x^2 instead of 32x, then the function is even; it DOES have a local maximum of 400 at x=4; and, since it is an even function, another local maximum is 400 at x=-4.
So your post should have started like this:
"Given G(x) = -x^4 + 32x^2 + 144, ...."
With this start, the problem is straightforward:
It is an even function because all terms have the variable to even powers.
Since it is an even function, if one local maximum is 400 at x=4, then another local maximum is 400 at x=-4.
Answer by math_tutor2020(3838)  (Show Source):
You can put this solution on YOUR website!
Part (a)
You can use the method tutor greenestamps mentioned. - If all exponents of a polynomial function are even, then the function is even.
- If all exponents of a polynomial function are odd, then the function is odd.
- If the polynomial function has a mix of even and odd exponents, then the function is neither even nor odd.
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 the function is neither even nor odd
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Another approach for part (a)
Recall that:- If g(x) = g(-x) for all x in the domain, then g(x) is even.
- If g(-x) = -g(x) for all x in the domain, then g(x) is odd.
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 g(x) is neither even nor odd
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 GeoGebra, but Desmos 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.
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.
The highest point on this curve is at (2,192)
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 this article
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).
Answer by ikleyn(53937)  (Show Source):
You can put this solution on YOUR website! .
After reading the analysis by @greenestamps, I am horrified and I am shocked,
to what extent is irresponsible and careless writing of this visitor.
No one can trust any single word or any single formula or any single number in his posts . . .
I have an advice to this visitor: consider to visit lessons/classes on Math reading and writing.
It is absolutely necessary to you, because your reading/writing/understanding skills in Math
are at the zero level (if not significantly below it . . . )
Couple more such posts, and I will write a message to the managers of this project
asking them to remove this person from this forum for his permanent inability
to provide a proper input in each and every his post.
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