Answer :
To determine which values of [tex]\( q \)[/tex] to check in order to minimize the cost function [tex]\( C(q) = 0.1 q^3 - 14.85 q^2 + 688.2 q + 300 \)[/tex] on the interval [tex]\( 30 \leq q \leq 70 \)[/tex], we follow these steps:
1. Find the derivative of [tex]\( C(q) \)[/tex]:
The first step is to calculate the derivative of the cost function, [tex]\( C'(q) \)[/tex]. This helps identify the critical points where the slope is zero.
2. Solve for critical points:
Set the derivative [tex]\( C'(q) \)[/tex] equal to zero and solve for [tex]\( q \)[/tex]. These solutions are the critical points where the function may have a minimum or maximum.
3. Check the critical points within the interval:
After finding the critical points, we need to check which of these points lie within the given interval [tex]\( 30 \leq q \leq 70 \)[/tex].
4. Consider the endpoints of the interval:
The endpoints [tex]\( q = 30 \)[/tex] and [tex]\( q = 70 \)[/tex] must also be considered because the minimum could occur at these boundaries.
5. Evaluate the cost at these points:
To determine the minimum cost, evaluate the cost function [tex]\( C(q) \)[/tex] at all identified points: the critical points within the interval and the endpoints.
From following these steps, the values of [tex]\( q \)[/tex] that need to be checked are [tex]\( q = 30, 37, 62, \)[/tex] and [tex]\( 70 \)[/tex]. Therefore, the values to consider for finding the minimum cost are listed in option A: [tex]\( q=30,37,62,70 \)[/tex].
1. Find the derivative of [tex]\( C(q) \)[/tex]:
The first step is to calculate the derivative of the cost function, [tex]\( C'(q) \)[/tex]. This helps identify the critical points where the slope is zero.
2. Solve for critical points:
Set the derivative [tex]\( C'(q) \)[/tex] equal to zero and solve for [tex]\( q \)[/tex]. These solutions are the critical points where the function may have a minimum or maximum.
3. Check the critical points within the interval:
After finding the critical points, we need to check which of these points lie within the given interval [tex]\( 30 \leq q \leq 70 \)[/tex].
4. Consider the endpoints of the interval:
The endpoints [tex]\( q = 30 \)[/tex] and [tex]\( q = 70 \)[/tex] must also be considered because the minimum could occur at these boundaries.
5. Evaluate the cost at these points:
To determine the minimum cost, evaluate the cost function [tex]\( C(q) \)[/tex] at all identified points: the critical points within the interval and the endpoints.
From following these steps, the values of [tex]\( q \)[/tex] that need to be checked are [tex]\( q = 30, 37, 62, \)[/tex] and [tex]\( 70 \)[/tex]. Therefore, the values to consider for finding the minimum cost are listed in option A: [tex]\( q=30,37,62,70 \)[/tex].
