Question

the graph of a companys profit p(t) in dollars, at month t is shown. complete parts a through e below.
a. \\(\frac{dp}{dt}>0\\) and \\(\frac{d^{2}p}{dt^{2}} = 0\\) at t = (type a whole number. use a comma to separate answers as needed.)

Explanation:

Step1: Recall derivative meaning

The first - derivative $\frac{dP}{dt}$ represents the rate of change of profit with respect to time. $\frac{dP}{dt}>0$ means the profit is increasing. The second - derivative $\frac{d^{2}P}{dt^{2}}$ represents the rate of change of the first - derivative, i.e., the concavity of the profit function. $\frac{d^{2}P}{dt^{2}} = 0$ indicates an inflection point.

Step2: Analyze the graph

We look for the points on the graph of $P(t)$ where the function is increasing (the slope of the tangent line is positive) and where the concavity changes (inflection point). By observing the graph of $P(t)$, we find the intervals where the function is increasing and the inflection points.
We can see from the graph that the function $P(t)$ is increasing ( $\frac{dP}{dt}>0$) and has an inflection point ( $\frac{d^{2}P}{dt^{2}} = 0$) around $t = 24$.

Answer:

24