Bapilievolia0o0

2023-02-18

Where does the graph of $y=\left(5{x}^{4}\right)-\left({x}^{5}\right)$ have an inflection point?

Dakota George

An inflection point on a graph is a point where the concavity changes. We'll look at the sign of the second derivative to investigate concavity:
$y=5{x}^{4}-{x}^{5}$
$y\prime =20{x}^{3}-5{x}^{4}$
$y\prime \prime =60{x}^{2}-20{x}^{3}=20{x}^{2}\left(3-x\right)$
Obviously $20{x}^{2}$ is always positive, so the sign of y'' is the same as the sign of $3-x$.
Which is positive for $x<3$ and negative for $x>3$. At $x=3$ the concavity changes.
Because an inflection point is a point on a graph, we need:
when $x=3$, we get
$y=5\left({3}^{4}\right)-{3}^{5}=5\left({3}^{4}\right)-3\left({3}^{4}\right)=2\left({3}^{4}\right)=2\left(81\right)=162$
The point (3, 162) in the only inflection point for the graph of $y=5{x}^{4}-{x}^{5}$

Do you have a similar question?