# How do you determine the intervals on which function is concave up/down & find points of inflection for y=4x^5-5x^4?

Oct 23, 2015

Investigate the sign of the second derivative.

#### Explanation:

$y = 4 {x}^{5} - 5 {x}^{4}$

$y ' = 20 {x}^{4} - 20 {x}^{3} = 20 \left({x}^{4} - {x}^{3}\right)$

$y ' ' = 20 \left(4 {x}^{3} - 3 {x}^{2}\right) = 20 {x}^{2} \left(4 x - 3\right)$

$y ' '$ is never undefined and $y ' ' = 0$ at $x = 0$ and at $x = \frac{3}{4}$.

Because $20 {x}^{2}$ is always positive, the sign of $y ' '$ is the same as the sign of $4 x - 3$ (or build a sign table of sign diagram or whatever you have learned to call it, for $y ' '$).

$y ' '$ is negative (so the graph of the function is concave down, for $x < \frac{3}{4}$ and

$y ' '$ is posttive (so the graph of the function is concave up, for $x > \frac{3}{4}$

The curve is concave down on the interval $\left(- \infty , \frac{3}{4}\right)$ and
concave up on $\left(\frac{3}{4} , \infty\right)$.

The int $\left(\frac{3}{4} , - \frac{81}{128}\right)$ is the only inflection point.