Intervals on which function is increasing and decreasingLet 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 &lt; q 5 4 and strictly decreasing for − q 5 4 &lt; x &lt; q 5 4 and strictly increasing again for x &gt; q 5 4 .

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Intervals on which function is increasing and decreasingLet   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  &amp;lt;            q              5      4       and strictly decreasing for   −            q              5      4        &amp;lt;  x  &amp;lt;            q              5      4       and strictly increasing again for   x  &amp;gt;            q              5      4      .

indivisast7 · Accepted Answer

Step 1Assuming we have   q  &amp;gt;  0, the derivative is positive whenever   5      x    4    &amp;gt;      q    2  , or equivalently       5        x    2    &amp;gt;  q.This can happen two ways, either with       5    4    x  &amp;gt;      q   if x is positive, or   −      5    4    x  &amp;gt;      q   if x is negative.Step 2For a negative derivative, the inequalities are reversed.I am not sure what you mean by &quot;contains a prime number&quot; - 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.

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.

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