Use derivatives to determine whether $\mathrm{f}$ is increasing or decreasing on the given interval. Use $\mathrm{L}_{4}$ or $R_{4}$, whichever is appropriate, to give an overestimate of the signed area on the given interval.
\[
f(x)=4 e^{x^{2}} \text { on }[0,2]
\]

Step 1 ：First, we need to find the derivative of the function to determine whether the function is increasing or decreasing on the given interval. The derivative of a function can tell us whether the function is increasing or decreasing. If the derivative is positive, the function is increasing. If the derivative is negative, the function is decreasing.

Step 2 ：The derivative of the function \(f(x)=4e^{x^{2}}\) is \(f'(x)=8x e^{x^{2}}\).

Step 3 ：Evaluating the derivative at the endpoints of the interval [0,2], we find that \(f'(0)=0\) and \(f'(2)=16e^{4}\). Since both of these values are non-negative, the function is increasing on the interval [0,2].

Step 4 ：Next, we need to use the Right Hand Rule (RHR) to estimate the area under the curve. The RHR is a method for approximating the definite integral of a function. It works by dividing the area under the curve into rectangles and then adding up the areas of these rectangles. The RHR uses the right endpoints of the subintervals.

Step 5 ：In this case, since we are given the interval [0,2], we can use the RHR. Since the function is increasing on this interval, the RHR will give an overestimate of the area under the curve.

Step 6 ：We divide the interval [0,2] into 4 subintervals, each of width \(\Delta x = 0.5\). The right endpoints of these subintervals are 0.5, 1.0, 1.5, and 2.0.

Step 7 ：Using the RHR, we find that the overestimate of the area under the curve is approximately 136.176.

Step 8 ：Final Answer: The function \(f(x)=4e^{x^{2}}\) is \(\boxed{increasing}\) on the interval [0,2]. The R4 overestimate of the signed area under the curve on this interval is approximately \(\boxed{136.176}\).