I’m confused by the difference between using f’(x) and f”(x) to find the minima and maxima of a curve

Imagine you have an object moving through time, like a car on the road. The distance that car travels=xThe speed of that object=f’(x)This is because speed is defined as the the rate of change of distance (speed=distance/time), so this is used to find the gradient of a graph, as the gradient is the measure of the change in value of a quantity. Therefore, for a curve the maxima or minima have a gradient that is equal to zero (since the graph is neither going up or down). This can help you find the points on a graph at which a maxima or minima can occur.Now, in this case f”(x) is the rate of change in speed of the car- which is also known as the acceleration. By differentiating f’(x) again to get f”(x), you can tell if the point is a minima or maxima:if it is greater than 0, then it’s a minima.if it is less than 0, then it’s a maxima.