Calculus project 6: Discovering derivatives

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I see you've provided a step-by-step solution to finding the derivative and extreme values of a function. Here's a cleaned-up version with explanations: ## Step 1: Find the Derivative To find the derivative of f(X) = 4X^3 - 320X^2 + 6000x, we'll use the formula: \[ f'(X) = rn^{(r-1)} \] where n is the coefficient of X and r is the power of X. In this case, n = 4 and r = 3. ## Step 2: Calculate the Derivative Plugging in the values, we get: \[ f'(X) = 12X^2 - 640X + 6000 \] ## Step 3: Find Critical Points To find critical points, we need to set f'(X) equal to 0 and solve for X. Let's try it the long way: ## Step 4: Solve the Quadratic Equation We have: \[ 12X^2 - 640X + 6000 = 0 \] We can factor out the GCF (4): \[ 4(1500 - 160X + 3X^2) = 0 \] Ignoring the factor 4, we get: \[ 1500 - 160X + 3X^2 = 0 \] Dividing all terms by 3, we get: \[ 500 - 53.33333333X + X^2 = 0 \] ## Step 5: Complete the Square Moving the constant term to the right, we get: \[ 500 + -53.33333333X + -500 + X^2 = 0 + -500 \] Reordering terms and combining like terms, we get: \[ -53.33333333X + X^2 = -500 \] ## Step 6: Add Half the Coefficient to Both Sides Taking half of the coefficient (-26.66666667) and squaring it (711.1111113), we add both sides: \[ -53.33333333X + 711.1111113 + X^2 = -500 + 711.1111113 \] Reordering terms and combining like terms, we get: \[ 711.1111113 + -53.33333333X + X^2 = 211.1111113 \] ## Step 7: Factor a Perfect Square Factoring the left side as a perfect square, we get: \[ (X - 26.66666667)^2 = 211.1111113 \] Taking the square root of both sides, we get: \[ |X - 26.66666667| = 14.529663152 \] ## Step 8: Solve for X Breaking this into two subproblems by setting (X - 26.66666667) equal to 14.529663152 and -14.529663152, we get: Subproblem 1: \[ X - 26.66666667 = 14.529663152 \] Solving for X, we get: \[ X = 41.196329822 \] Subproblem 2: \[ X - 26.66666667 = -14.529663152 \] Solving for X, we get: \[ X = 12.137003518 \] ## Step 9: Set X equal to Given Values Since we know that anything above 30 will result in a negative number, we'll use X = 12.137003518. The final answer is: $\boxed{32832}$