Video Transcript
The graph of a function 𝑦 equals
𝑓 of 𝑥 is shown, at which point are d𝑦 by d𝑥 and d two 𝑦 by d𝑥 squared both
negative.
So we’ve been given the graph of a
function itself. And we’re asked to use this to
determine at which of these five points both the first and second derivatives of the
function are negative. First, let’s consider the sign of
the first derivative d𝑦 by d𝑥 at each point. Recall that the first derivative of
a function at a point gives the slope of the tangent to the curve at that point. So by sketching in tangents to the
curve at each point, we can determine the sign of their first derivative.
We see that at point 𝐴 the tangent
is sloping downwards. So the first derivative, d𝑦 by
d𝑥, is indeed negative at point 𝐴. However, at points 𝐵 and 𝐶, the
tangents are each sloping upwards, which tells us that the first derivative d𝑦 by
d𝑥 will be positive at both 𝐵 and 𝐶. At point 𝐷, the tangent to the
curve is horizontal. So the first derivative will be
equal to zero, not negative at this point. Finally, at point 𝐸, we see that
the tangent is sloping downwards. So the first derivative will also
be negative at point 𝐸. We’re, therefore, left with only
two options 𝐴 and 𝐸. Next, we need to consider the sign
of the second derivative at each of these points. And this is linked to the concavity
of the curve at each point.
Recall that the curve is said to be
concave downward on a particular interval, if the tangents to the curve in that
interval lie above the curve itself. We can also see that when a curve
is concave downwards, the slope of its tangent is decreasing. And therefore, its first derivative
is also decreasing. When a function is decreasing, then
its derivative is negative. And as the derivative of the first
derivative is the second derivative, it follows that d two 𝑦 by d𝑥 squared will be
less than zero when a curve is concave downward.
This isn’t required here, but a
curve is said to be concave upward when the reverse is true. The tangents to the graph lie below
the graph itself. The first derivative is increasing
and therefore, the second derivative is positive. By considering the graph of 𝑓 of
𝑥, we can see that the tangent we drew at point 𝐸 lies above the curve. And indeed, the shape of the curve
is concave downward in this region. However, if we look at point 𝐴,
the tangent we drew here is below the curve. And so the graph is concave upward
at point 𝐴. This tells us then the second
derivative will be negative at point 𝐸, whereas it will be positive at point
𝐴.
We’re left then with only one point
at which both the first and second derivatives are negative. It’s point 𝐸. We’ve seen in this example how to
determine something about the first and second derivatives of a function from a
graph of the function itself.
