Cost Function

Question

Total cost function (TC) of a product is TC=1000+30q-25q²-4q³ where q = quantity

How to find following

• Marginal Cost
• At what level of q AVG is minimum
• Average cost function

Answer 1

Here is a basic algorithm using the newton raphson method to find derivates:

#include <stdio.h>

#include <math.h>

#include <stdlib.h>

static inline double TotalCost(const double q) {

return 1000.0 + 30.0*q - 25.0*q*q - 4.0*q*q*q;

}

// Marginal Cost (derivative of TC)

static inline double MarginalCost(const double q) {

return 30.0 - 50.0*q - 12.0*q*q;

}

static inline double AverageCost(const double q) {

if (q <= 0.0) return NAN;

return TotalCost(q) / q;

}

// Derivative of Average Cost (to find minimum)

static inline double dAC(const double q) {

if (q <= 0.0) return NAN;

return -1000.0/(q*q) - 25.0 - 8.0*q;

}

// Second derivative of AC (for Newton's method)

static inline double d2AC(const double q) {

if (q <= 0.0) return NAN;

return 2000.0/(q*q*q) - 8.0;

}

// Newton-Raphson to find root of dAC(q)=0

// Returns root if found and positive, otherwise NAN

double findMinAC(const double initial_guess, const int max_iter, const double tol) {

double q = initial_guess;

for (int i = 0; i < max_iter; i++) {

double f = dAC(q);

if (fabs(f) < tol) return q;

double df = d2AC(q);

if (fabs(df) < 1e-12) break; // avoid division by zero

double q_new = q - f / df;

if (q_new <= 0.0) break; // only positive quantities are meaningful

q = q_new;

}

return NAN;

}

int main(void) {

double q;

printf("Enter quantity q (positive): ");

if (scanf("%lf", &q) != 1 || q <= 0.0) {

printf("Invalid input. Please enter a positive number.\n");

return EXIT_FAILURE;

}

printf("\n--- Results for q = %.4f ---\n", q);

printf("Total Cost (TC) = %.4f\n", TotalCost(q));

printf("Marginal Cost (MC) = %.4f\n", MarginalCost(q));

printf("Average Cost (AC) = %.4f\n", AverageCost(q));

printf("Derivative of AC = %.4f\n", dAC(q));

return EXIT_SUCCESS;

}