The expression x to the power of 2 - 2mx has a minimum value as x varies Find the minimum value of x to the power of 2 - 2mx and Give your answer in terms of m?

Answer 1

If this is a homework question, please consider answering it on your own first, otherwise the value of the homework assignment, i.e. to reinforce the lesson, will be lost to you.

The minimum or maximum of a function y = f(x) can be found when the deriviative dy/dx f(x), or slope, is zero. In the case of a polynomial of degree two, which this question is, there is only one point where the deriviative is zero. In the general case of a polynomial of degree three or higher, however, any point where the slope is zero might only be a local minimum or maximum, so care must be taken. Also, you must always consider the second deriviative, in order to verify if that point is a minumum or a maximum.

The first deriviative of y = x2 - 2mx is 2x - 2m. (Remember, m is constant)

By simple algebra, the solution of 2x - 2m = 0 is x = m.

To determine if this is a minimum or a maximum, take the second deriviative.

The second deriviative of y = x2 - 2mx is the first deriviative of y = 2x - 2m, which is 2.

Since 2 is always positive, the slope is always increasing, so the point x = m is a minimum.