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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?
Wiki User
∙ 14y ago
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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.
Wiki User
∙ 14y ago
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