The graph of the derivative 𝑓 prime of a function 𝑓 is shown. At what values of 𝑥 does 𝑓 have a local maximum or minimum?
Let’s, firstly, recall what it means for a point to be a local maximum or minimum. At maximum and minimum points, our slope is equal to zero. Remember, the slope function is the derivative. So we’re looking for points where the derivative is zero. Well, we can see from our graph of 𝑓 prime that the derivative is equal to zero at 𝑥 equals one and also at 𝑥 equals five. Great, but how do we know if these points are local maximums or local minimums. Well, on the graph of 𝑓, the slope to the left of the maximum point is positive. And to the right of the maximum point, the slope is negative. And for a minimum point, the slope to the left of the minimum point is negative and the slope to the right of the minimum point is positive.
So here, on the graph of 𝑓 prime at 𝑥 equals one, we can see that the derivative goes from being in negative values of 𝑓 prime of 𝑥 to positive values of 𝑓 prime of 𝑥. The derivative goes from negative to positive. So this must be a local minimum. At 𝑥 equals five, we can see that the derivative goes from positive to negative. So this must be a local maximum. And so we can conclude that 𝑓 has a local maximum at 𝑥 equals five and a local minimum at 𝑥 equals one.