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Find all local minimums and maximums of the function f(x)=x^{2}.

Local minimums and maximums may occur never, once, or many times depending on the function. However, they occur only at critical points, that is only at points where the function is defined and the derivative is 0 or undefined. The derivative of our function is


This derivative is always defined and is equal to 0 only when x=0. So our only critical point is x=0. To decide if this is a local maximum or minimum let us look at the behavior of the derivative to the left and to the right of it. Since this is the only point where the derivative is ever equal to 0 it must be true that to the left of x=0 the derivative is always negative or always positive and the same can be said to the right of x=0. If the derivative is positive that tells us that the function is increasing in that region and if the derivative is negative that tells us that the function is decreasing in that region. This information will tell us if the critical point is a local max, a local min, or neither.

Since the behavior is the same for any point left of x=0 and also always the same for any point right of x=0, just pick one point on each side and find the value of the derivative at that x-value.



Since the derivative to the left of x=0 is negative, that tells us that the function is decreasing up to that point and since the derivative is positive to the right, we know the function is increasing on the interval (0,\infty ). If the function goes from decreasing to increasing, this must be a local minimum at the point (0,0) which is easily verified by looking at a graph.

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