If the cost function is C(x) = 3x^2 + 12x + 5, find the minimum cost. (2020)

If the cost function is C(x) = 3x^2 + 12x + 5, find the minimum cost. (2020)

Step 1: Identify the cost function, which is C(x) = 3x^2 + 12x + 5.
Step 2: Recognize that this is a quadratic function in the form of ax^2 + bx + c, where a = 3, b = 12, and c = 5.
Step 3: To find the minimum cost, use the formula for the x-coordinate of the vertex of a parabola, which is x = -b/(2a).
Step 4: Substitute the values of a and b into the formula: x = -12/(2*3).
Step 5: Calculate the denominator: 2*3 = 6.
Step 6: Now calculate x: x = -12/6 = -2.
Step 7: To find the minimum cost, substitute x = -2 back into the cost function C(x).
Step 8: Calculate C(-2): C(-2) = 3(-2)^2 + 12(-2) + 5.
Step 9: Calculate (-2)^2 = 4, so C(-2) = 3*4 + 12*(-2) + 5.
Step 10: Now calculate: 3*4 = 12, and 12*(-2) = -24.
Step 11: Combine the results: C(-2) = 12 - 24 + 5.
Step 12: Finally, calculate: 12 - 24 = -12, and -12 + 5 = -7.

Quadratic Functions – Understanding the properties of quadratic functions, including how to find their minimum or maximum values using the vertex formula.
Cost Functions – Applying mathematical concepts to real-world scenarios, specifically in determining minimum costs based on a given cost function.