Question

In Exercises \(7-12\) , use the Concavity Test to determine the intervals on which the graph of the function is (a) concave up and (b) concave down. $$y=e^{x}\( \)0 \leq x \leq 2 \pi$$

Short answer

The function \( y = e^x \) is concave up on the interval \( 0 \leq x \leq 2 \pi \) and never concave down.

Step-by-step solution

Calculate the first derivative

The first derivative of the function \( y = e^x \) is \( y' = \frac{d}{dx}(e^x) = e^x \).

Calculate the second derivative

The second derivative of the function is \( y'' = \frac{d}{dx}(e^x) = e^x \).

Determine where the second derivative is positive or negative

Since \( y'' = e^x \) and \( e^x \) is always greater than 0 for all real x, we can say that the second derivative is always positive. Therefore, the graph of the function is concave up on the entire interval \( 0 \leq x \leq 2 \pi \). And since it’s always concave up, it’s never concave down.