Help me conclude a statement about two points given the following....

Question

Two points on the graph of $y=kx^p$ are labeled $A$ and $C$. Point $A$ has coordinates $(a,b)$, where $0<a<1$ and point $C$ has coordinates $(c,d)$, where $1<c$.

If we are told that that the product $k\cdot p$ is negative, what can be concluded about the points $A$ and $C$? Defend your answer.

I can't figure this out. I have tried multiple times. The hint in the book that I have to prove y is decreasing so I figured then $a<b$ and $c>d$. How would I prove this.

Answer 1

Note that $y' = kpx^{p-1}$ so, for $x > 0$, $y'< 0$.

Therefore $y$ is decreasing for $x > 0$.

Since $0 < a < 1 < c$, $y$ is decreasing at both $A$ and $C$.

Since $a < 1 < c$, $A$ is to the left of $C$. Since $y$ is decreasing, $A$ is above $C$.

Answer 2

Note first that $0<a<1<c$, and keep that in mind. Note also that $b=ka^p$ and $d=kc^p$, so that $A$ is the point $(a,ka^p)$, and $C$ is the point $(c,kc^p)$.

If $pk<0$, then one of $p$ and $k$ is positive, and the other is negative. Consider the two cases separately.

• Suppose that $p>0$ and $k<0$. You should know that when $p>0$, the function $f(x)=x^p$ is increasing on the positive reals; that is, if $0<x_0<x_1$, then $0<x_0^p<x_1^p$. Since $0<a<c$, that means that $a^p<c^p$. But you’re not interested in $a^p$ and $c^p$: you want $ka^p$ and $kc^p$, where $k$ is negative. When you multiply an inequality by a negative number, you must reverse the direction of the inequality, so $ka^p>kc^p$. In other words, the point $C$ is to the right of $A$, because $c>a$, but it’s also below $A$, because $d=kc^p<ka^p=b$. To get from $A$ to $C$, you must go to the right and down, so the graph of $y=kx^p$ has dropped from $A$ to $C$: the function is decreasing.

• Now suppose that $p<0$ and $k>0$. You should know that when $p<0$, the function $f(x)=x^p$ is decreasing on the positive reals: if $0<x_0<x_1$, then $x_0^p>x_1^p$. (If you’re not sure, draw graphs for a few negative values of $p$.) Since $a<c$, this means that $a^p>c^p$. Multiplying an inequality by a positive number doesn’t change the direction of the inequality, so what can you say about $ka^p$ and $kc^p$? And what does that tell you about the relative positions of $A$ and $C$?