Suppose that f is a function and we want to determine if f has a maximum or minimum at x. If x is a maximum of f, then f is increasing to the left of x and decreasing to the right of x. Similarly, if x is a minimum of f, then f is decreasing to the left of x and increasing to the right of x. If f is increasing on both sides of x, or if f is decreasing on both sides of x, then x is not a maximum or a minimum.
If f is differentiable in a neighbourhood of x, we can rephrase the conditions of being increasing or decreasing in terms of the derivative of f. When the derivative of f is positive, then f is increasing, and when the derivative of f is negative, then f is decreasing. The first derivative test now states:
If there exists a positive number r such that f' is continuous between x-r and x+r, and for every y such that x-r<y<x we have f'(y)>0, and for every y such that x<y<x+r we have f'(y)<0, then f has a maximum at x.
If there exists a positive number r such that f' is continuous between x-r and x+r, and for every y such that x-r<y<x we have f'(y)<0, and for every y such that x<y<x+r we have f'(y)>0, then f has a minimum at x.
If there exists a positive number r such that f' is continuous between x-r and x+r, and for every y such that x-r<y<x or x<y<x+r we have either f'(y)>0 or f'(y)<0, then f has neither a maximum nor a minimum at x.
If f' is not continuous between x-r and x+r for any r, or if none of the above conditions hold for any r for which f' is continuous between x-r and x+r, then the test fails.
In this case, the Second DerivativeTest can still be used to characterize critical points, by considering the eigenvalues of the Hessian matrix of second partial derivatives of the function at the critical point.
The common thread is that the derivative at a point serves as a linear approximation of the function at that point.
Perhaps the most natural situation is that of functions between differentiable manifolds; the derivative at a certain point then becomes a linear transformation between the corresponding tangent spaces and the derivative function becomes a map between the tangent bundles.
In calculus, a branch of mathematics, the firstderivativetest determines whether a given critical point of a function is a maximum, a minimum, or neither.
If f is differentiable in a neighbourhood of x, we can rephrase the conditions of being increasing or decreasing in terms of the derivative of f.
When the derivative of f is positive, then f is increasing, and when the derivative of f is negative, then f is decreasing.
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