Let p(x)=x^5-q^2 x-q, where q is a prime number. I want to understand how to determine when the function will be decreasing and increasing on the intervals given below. We compute p′(x)=5x^4-q^2 and look for the critical points.

Amina Richards 2022-10-11 Answered
Intervals on which function is increasing and decreasing
Let p ( x ) = x 5 q 2 x q, where q is a prime number. I want to understand how to determine when the function will be decreasing and increasing on the intervals given below.
We compute p ( x ) = 5 x 4 q 2 and look for the critical points.
5 x 4 q 2 = 0 x = ± q 5 4
Hence we have to investigate the behavior of p′(x) for each of these intervals ( , q 5 4 ), ( q 5 4 , q 5 4 ) and ( q 5 4 , ) this will indicate when the function will be increasing and decreasing. How can this be determined when the expression q 5 4 contains a prime number???
The answer should be : the function will be increasing for x < q 5 4 and strictly decreasing for q 5 4 < x < q 5 4 and strictly increasing again for x > q 5 4 .
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Answers (1)

indivisast7
Answered 2022-10-12 Author has 13 answers
Step 1
Assuming we have q > 0, the derivative is positive whenever 5 x 4 > q 2 , or equivalently 5 x 2 > q.
This can happen two ways, either with 5 4 x > q if x is positive, or 5 4 x > q if x is negative.
Step 2
For a negative derivative, the inequalities are reversed.
I am not sure what you mean by "contains a prime number" - the function is presumably being taken over the real numbers, over which the [positive real] square and fourth roots of non-negative real numbers are well-defined.
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