How to find the minimum cost of quadratic function?

Question

The total cost of producing a type of truck is given by c(x)=21000-40x+0.1x^2. Where x is the number of trucks produced. How many trucks should be produced to incur minimum cost?

Answer 1

The function's difference quotient is needed for this. Differentiating the expression for c(x) we get -40+0.2x. This the gradient of the graph of the function. When the gradient is zero, we have a maximum or minimum. The shape of this graph is a narrow U, so the gradient is zero at the minimum point. Solving -40+0.2x=0, we get x=40/0.2=200. So 200 trucks incur minimum cost.