SOLUTION: I am sorry I know this is a lengthy question but I am just not sure what the answers are. f(x)=-x^10+kx^4+7 i. What are the end behaviours of this type of function (what qua

Question 1119975: I am sorry I know this is a lengthy question but I am just not sure what the answers are.
f(x)=-x^10+kx^4+7
i. What are the end behaviours of this type of function (what quadrant does it begin and end in?)
ii. What is the maximum and minimum number of�x-intercepts for this type of function?
iii. What is the maximum and minimum number of turns for this type of function?
iv. State if there are any restrictions on the domain and range on this type of function.
Any help to any of these questions would be so helpful thank you so much in advance.

Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

I answered this one already.

When

is positive,

is positive, so when

gets very large, the function gets very large. When

is negative,

is positive, so when

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.

-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 distinct zeros exist for any real number value for

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:

and

for several values of

John

My calculator said it, I believe it, that settles it

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