Determine the number of critical points of the following graph.
We’ve been given the graph of some function 𝑦 equals 𝑓 of 𝑥. And we’ve been asked to determine the number of critical points of this graph. We recall that at the critical points of a function, its first derivative 𝑓 prime of 𝑥 is equal to zero or is undefined. This graph shows a continuous function. So we don’t need to be concerned about where its derivative may be undefined. Instead, we only need to consider points for which the first derivative is equal to zero.
We should recall that the first derivative of a function gives the slope of its graph. And the slope of its graph at any given point is the same as the slope of the tangent to the curve at that point. If the first derivative is equal to zero at a critical point, then the slope of the graph is also equal to zero at this point. This means that the tangent to the graph will be a horizontal line.
We therefore just need to consider on the figure how many points there are at which the tangent will be horizontal. There’s one point here and a second point here and also a third point here. We also notice that at each of these points, the sign of the slope of the curve changes from negative to positive at points one and three and positive to negative at point two. So we have two local minima and one local maxima which we may recognize from the shape of the curve.
Our answer then is that this graph has three critical points.