When f(x) changes from increasing to decreasing – maximum, decreasing to increasing – minimum. Also check endpoints if they are included.
Find the relative minimum/maximum value of f(x) given an equation.
To determine whether a function f(x) has a maximum or minimum at a point of change in slope, we use the first derivative test.
If we take the derivative of f(x) and set it equal to zero, we can find the critical points of the function, which are potential maxima or minima. If the second derivative at these critical points is negative, then we have a local maximum. If the second derivative is positive, then we have a local minimum.
If f(x) changes from increasing to decreasing at a point, then that point is a local maximum. If f(x) changes from decreasing to increasing at a point, then that point is a local minimum.
However, we also need to check the endpoints of the interval in question. For example, if we are looking at a function on the interval [a,b], we need to check whether a and b are also potential maximum or minimum points. To do this, we can simply substitute a and b into the function and compare the values.
It should be noted that having a local maximum or minimum does not necessarily mean that the function has an absolute maximum or minimum on the interval. It is possible that there is a higher or lower point outside of the interval.
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