Final answer:
To find the maximum value of the function 8x² − e⁴X within the range of 0 to 8, one must compute the derivative, set it to zero to find critical points, and then assess the values at these points alongside the interval's endpoints. The information provided, however, describes a different quadratic equation and cannot be used for this problem.
Step-by-step explanation:
To find the maximum value of the expression 8x² − e⁴X within the horizontal span of 0 to 8, we can use calculus to find the derivative of the expression and determine where it is zero, since the derivative tells us where the function's slope changes sign, which indicates a local maximum or minimum.
First, we take the derivative of the function with respect to x:
f'(x) = d/dx(8x² − e⁴X) = 16x - e⁴X*4.
Next, we set the derivative equal to zero and solve for x:
16x - 4e⁴X = 0.
This will give us the critical points where the function reaches its maximum or minimum within the range.
We would then examine the critical points and the endpoints of the interval (0 and 8) by plugging them back into the original function to see which yields the highest value. Since the given expression involves exponential and quadratic terms, numerical methods or a graphing calculator might be required to accurately find this maximum value within the given domain.
However, the information provided seems inconsistent with the problem posed since it talks about a different quadratic equation unrelated to the initial exponential-quadratic expression. Therefore, we cannot proceed with the calculation using the given constants (a = 4.90, b = 14.3, and c = -20.0), as they do not apply to the expression 8x² − e⁴X. Instead, we should use appropriate calculus methods to analyze the correct expression.