Sure, let’s go through the solution step by step.
Given the revenue function [tex]\( R(x) = -x^3 + 57x^2 + 1000 \)[/tex] for [tex]\( 0 \leq x \leq 24 \)[/tex]:
### (a) Finding [tex]\( R''(x) \)[/tex]:
1. First Derivative [tex]\( R'(x) \)[/tex]:
To find the second derivative, we first find the first derivative [tex]\( R'(x) \)[/tex].
[tex]\[
R'(x) = \frac{d}{dx}(-x^3 + 57x^2 + 1000)
\][/tex]
We apply the power rule of differentiation:
[tex]\[
R'(x) = -3x^2 + 114x
\][/tex]
2. Second Derivative [tex]\( R''(x) \)[/tex]:
Next, we find the second derivative [tex]\( R''(x) \)[/tex] by differentiating [tex]\( R'(x) \)[/tex] again.
[tex]\[
R''(x) = \frac{d}{dx}(-3x^2 + 114x)
\][/tex]
Applying the power rule again:
[tex]\[
R''(x) = -6x + 114
\][/tex]
So, [tex]\[ R''(x) = -6x + 114 \][/tex]
### (b) Finding the Point of Diminishing Returns:
The point of diminishing returns occurs where the second derivative [tex]\( R''(x) \)[/tex] equals zero. This indicates a change in the concavity of the function.
1. Set [tex]\( R''(x) = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[
-6x + 114 = 0
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
-6x = -114
\][/tex]
[tex]\[
x = 19
\][/tex]
2. Finding [tex]\( R(19) \)[/tex]:
To find the corresponding revenue at this point, we substitute [tex]\( x = 19 \)[/tex] back into the original revenue function [tex]\( R(x) \)[/tex].
[tex]\[
R(19) = -(19)^3 + 57 (19)^2 + 1000
\][/tex]
So, the ordered pair at the point of diminishing returns is [tex]\( (19, R(19)) \)[/tex], but since the problem asks just for the point [tex]\( x \)[/tex], we have our answer as [tex]\( x = 19 \)[/tex].
### Summary of Results:
(a) [tex]\( R''(x) = -6x + 114 \)[/tex]
(b) The point of diminishing returns is [tex]\((19)\)[/tex].
