Let's carefully solve each part of the question.

Given:

1. $f(x) = e^x \sin x$ on the interval $[-\pi, \pi]$.
2. The derivatives provided are:
   • $f'(x) = e^x (\sin x + \cos x)$
   • $f''(x) = 2e^x \cos x$.

Part (a): Where is $f(x)$ increasing?

To determine where $f(x)$ is increasing, we need to analyze where $f'(x) > 0$ (the first derivative is positive).

From the given derivative: $f'(x) = e^x (\sin x + \cos x).$

Steps:

1. Factor $e^x$: Since $e^x > 0$ for all real $x$, the sign of $f'(x)$ depends only on $\sin x + \cos x$.

2. Solve $\sin x + \cos x > 0$: To solve this inequality, notice: $\sin x + \cos x = \sqrt{2} \sin \left( x + \frac{\pi}{4} \right).$ The sign of $\sin \left( x + \frac{\pi}{4} \right)$ determines where $\sin x + \cos x$ is positive.

   • $\sin \theta > 0$ when $\theta \in (0, \pi)$ within a $2\pi$-period.

   Therefore: $x + \frac{\pi}{4} \in (0, \pi) \implies x \in \left( -\frac{\pi}{4}, \frac{3\pi}{4} \right).$

3. Final Answer for Part (a): $f(x)$ is increasing on the interval: $\left( -\frac{\pi}{4}, \frac{3\pi}{4} \right).$

Part (b): Where is $f(x)$ concave up?

To determine where $f(x)$ is concave up, we analyze where $f''(x) > 0$ (the second derivative is positive).

From the given second derivative: $f''(x) = 2e^x \cos x.$

Steps:

1. Factor $2e^x$: Since $2e^x > 0$ for all $x$, the sign of $f''(x)$ depends only on $\cos x$.

2. Solve $\cos x > 0$:

   • $\cos x > 0$ when $x \in (-\frac{\pi}{2}, \frac{\pi}{2})$ within the interval $[-\pi, \pi]$.

3. Final Answer for Part (b): $f(x)$ is concave up on the interval: $\left( -\frac{\pi}{2}, \frac{\pi}{2} \right).$

Final Results:

1. (a) $f(x)$ is increasing on $\left( -\frac{\pi}{4}, \frac{3\pi}{4} \right)$.
2. (b) $f(x)$ is concave up on $\left( -\frac{\pi}{2}, \frac{\pi}{2} \right)$.

Do you need further clarification or a detailed graph of these intervals?

Here are 5 related questions you might consider:

1. How do you determine the intervals where $f(x)$ is decreasing?
2. What is the behavior of $f(x)$ at the critical points?
3. How do you graph $f(x) = e^x \sin x$ and identify its key features?
4. How can you use the first derivative test to confirm local maxima or minima?
5. How is concavity related to the inflection points of $f(x)$?

Tip: To determine where a function is increasing, always analyze the sign of the first derivative!