Differential Calculus. We have a curve describing the variable Y as some function
of the variable X:
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Various straight lines are drawn tangent to the curve Y = f(X). Only one of these can lie flat, i.e., have a slope of zero, the one shown here as b=0. Any line tangent to the left or right of that (X=i, X=k) will connect with higher values of Y. So at the value X=j, Y is at a minimum.
This is obvious from the graph. If we have the equation for Y we can also determine what the minimum is, provided there is one. For any curve described by the equation
where a is a constant and n is an exponent, the slope of a tangent line, called the "derivative", will be given by![]()
There are other rules (for logarithms and trigonometric functions, e.g.) presented in any elementary calculus text, but they need not concern us here.![]()
If the original equation contains several terms, differentiation is performed on each term separately:
Once we have the derivative we can set it equal to zero, then solve to identify the value of X which minimizes Y.![]()
Technically, the derivative (slope of the tangent line) equal to zero applies when Y is at a minimum or a maximum (and also at what is called a point of inflection, not shown):
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If the second derivative is positive (the change in slope is toward more positive values), then we have a minimum.