Question 1119976
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I answered this one already.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  f(x)\ =\ 7x^4\ +\ kx^3\ +\ 8]


When *[tex \Large x] is positive, *[tex \Large x^4] is positive, so when *[tex \Large x] gets very large, the function gets very large.  When *[tex \Large x] is negative, *[tex \Large x^4] is positive, so when *[tex \Large x] gets very small, the function gets very large.


The number of zeros of a polynomial function is equal to the degree of the function.  This includes both real and complex zeros and takes into account multiplicities. *[tex \Large x]-intercepts represent real zeros, so the maximum number of real zeros for any polynomial function is equal to the degree of the function.  Complex zeros always appear in conjugate pairs, thus the minimum number of real zeros for a polynomial function of ODD degree is one.  The minimum number of real zeros for a polynomial function of EVEN degree is zero.


Turning points occur where the first derivative is zero.  You need to take the first derivative of the function and determine how many <b><i>distinct</i></b> zeros exist for any real number value for *[tex \Large k]


This is a polynomial function, therefore there are no restrictions on the domain.


Since you have discovered by now that the end behavior on both ends is the same, i.e. the function increases without bound on either end, it is clear that there must be an absolute minimum for the function and this must occur at one of the possible turning points.  Investigate:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(0)]


and


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f\(\frac{-3k}{28}\)]


for several values of *[tex \Large k]
								
								
John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it
<img src="http://c0rk.blogs.com/gr0undzer0/darwin-fish.jpg">
*[tex \Large \ \
*[tex \LARGE \ \ \ \ \ \ \ \ \ \  
								
{{n}\choose{r}}

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