Question Video: Discussing the Differentiability of a Function at a Point from Its Graph Mathematics

Video Transcript

The figure shows the graph of 𝑓. What can be said about the differentiability of 𝑓 at 𝑥 equals negative four?

Here, we’ve been given a graph which is defined over the interval from negative seven to negative one. Through this interval, we see that our curve is smooth at all points, aside from the point where 𝑥 is equal to negative four. At this point of coordinates negative four, five, we have a sharp corner. This means that the slope of the tangent, just to the left of 𝑥 equals negative four, will be different to the slope of the tangent just to the right of 𝑥 equals negative four. Here, we can even go so far as to say that one of our slopes will be positive and one of our slopes will be negative. Given that we have two different tangents on either side, it follows that it is not possible to define a tangent at 𝑥 equals negative four. And therefore, it’s also not possible to define the derivative.

If we were to imagine the graph of 𝑦 equals 𝑓 dash of 𝑥 our first derivative, we would expect to see a sharp jump in the 𝑦-value when 𝑥 equals negative four. From our observations, we conclude that the function is not differentiable at 𝑥 equals negative four because the functions rate of change is different on both sides of that point. And with this statement, we have answered our question.