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.