### Video Transcript

The graph of a function ๐ฆ equals
๐ of ๐ฅ is shown, at which point are d๐ฆ by d๐ฅ and d two ๐ฆ by d๐ฅ squared both
negative.

So weโve been given the graph of a
function itself. And weโre asked to use this to
determine at which of these five points both the first and second derivatives of the
function are negative. First, letโs consider the sign of
the first derivative d๐ฆ by d๐ฅ 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 ๐ด the tangent
is sloping downwards. So the first derivative, d๐ฆ by
d๐ฅ, is indeed negative at point ๐ด. However, at points ๐ต and ๐ถ, the
tangents are each sloping upwards, which tells us that the first derivative d๐ฆ by
d๐ฅ will be positive at both ๐ต and ๐ถ. At point ๐ท, the tangent to the
curve is horizontal. So the first derivative will be
equal to zero, not negative at this point. Finally, at point ๐ธ, we see that
the tangent is sloping downwards. So the first derivative will also
be negative at point ๐ธ. Weโre, therefore, left with only
two options ๐ด and ๐ธ. 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 ๐ฆ by d๐ฅ squared will be
less than zero when a curve is concave downward.

This isnโt 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 ๐ of
๐ฅ, we can see that the tangent we drew at point ๐ธ lies above the curve. And indeed, the shape of the curve
is concave downward in this region. However, if we look at point ๐ด,
the tangent we drew here is below the curve. And so the graph is concave upward
at point ๐ด. This tells us then the second
derivative will be negative at point ๐ธ, whereas it will be positive at point
๐ด.

Weโre left then with only one point
at which both the first and second derivatives are negative. Itโs point ๐ธ. Weโve seen in this example how to
determine something about the first and second derivatives of a function from a
graph of the function itself.