need help with another problem don"t know much English

Question

A baker has modeled the monthly operating costs for making wedding cakes by the

function y = 1/2x^2 - 12x + 150 where y is the total cost in dollars and x is the number of cakes prepared. What is the minimum operating cost?

Answer 1

given function y = 1/2 x² - 12x + 150

it's a quadratic which is a parabola and to find the minimum we should differentiate to get

^dy/_dx = x - 12

and for minimum (or maximum) ^dy/_dx = gradient = 0

hence 0 = x -12 => x = 12

Strictly, we don't know if it's a minimum or maximum, but as there's only one, and the question asks about minimum, it's pretty safe to assume this. Or, you can find the second derivative (differentiate x - 12), which is 1. 2nd derivative is positive for a minimum and negative for a maximum.

plug the x value into the orginal equation to get y = 1/2 (12)² - 12(12) + 150

=> y = 72 - 144 + 150 = 78

The point (12, 78) is the vertex of the parabola.

Hence minimum operating cost is $78 (which occurs when 12 cakes made)

You can double-check with Desmos graphing or similar.