# How do you find and classify all the critical points and then use the second derivative to check your results given h(t)=-4.9t^2+39.2t+2?

Aug 24, 2016

You first need to find the critical points where $h ' \left(t\right) = 0$, and then check the sign of the second derivative in those points.

#### Explanation:

Let's calculate first the first derivative $h ' \left(t\right) = \left(- 4.9\right) \cdot 2 \cdot t + 39.2$

Hence,

$h ' \left(t\right) = 0$ means $- 9.8 t + 39.2 = 0$, and thus the only critical point is $t = 4$.

Let's calculate now the second derivative in $t = 4$. But the second derivative is $- 9.8$ anywhere, so in particular the second derivative in $t = 4$ is $- 9.8$

Since the second derivative is negative in the critical point, the function has a maximum at that point.