### Video Transcript

Consider the parametric curve π₯ is
equal to cos π and π¦ is equal to sin π. Determine whether this curve is
concave up, down, or neither at π is equal to π by six.

Here, we have been asked about the
concavity of a curve. We know that, in order to determine
the concavity of a curve, we need to consider the second derivative of π¦ with
respect to π₯. We know that when d two π¦ by dπ₯
squared is greater than zero, our curve is concave down. And when itβs less than zero, our
curve is concave up. From this, we can see that, in
order to consider the concavity of our curve, we first need to find d two π¦ by dπ₯
squared. Now, weβve been given our curve in
terms of parametric equations. Therefore, we can use the following
formula to find d two π¦ by dπ₯ squared. This formula tells us that d two π¦
by dπ₯ squared is equal to d by dπ of dπ¦ by dπ₯ over dπ₯ by dπ. In order to use this, we first need
to find dπ¦ by dπ₯, which we also know a formula for. We have that dπ¦ by dπ₯ is equal to
dπ¦ by dπ over dπ₯ by dπ. Therefore, we can start by finding
dπ¦ by dπ and dπ₯ by dπ.

Weβve been given that π₯ is equal
to cos π. And π¦ is equal to sin π. Differentiating cos π with respect
to π, we obtain that dπ₯ by dπ is equal to negative sin π. And differentiating sin π with
respect to π, we obtain that dπ¦ by dπ is equal to cos π. And substituting these into our
equation for dπ¦ by dπ₯, we obtain that dπ¦ by dπ₯ is equal to negative cos π over
sin π or negative cot π. Next, we need to differentiate dπ¦
by dπ₯ with respect to π, which is equivalent to d by dπ of negative cot π. Now, we know that cot π
differentiates to give negative csc squared π. And so we can see that negative cot
π will differentiate to csc squared π. Now, we have found d by dπ of dπ¦
by dπ₯ and dπ₯ by dπ. And so we obtain that d two π¦ by
dπ₯ squared is equal to csc squared π over negative sin π. Since csc π is equal to one over
sin π, we have that d two π¦ by dπ₯ squared is equal to negative csc cubed π.

At this stage, we found d two π¦ by
dπ₯ squared. We just need to evaluate it at π
is equal to π by six. At π is equal to π by six, we
have that it is equal to negative csc cubed of π by six. And we can use the fact that csc is
equal to one over sin, giving us that this is equal to negative one over sin cubed
of π by six. We know that sin of π by six is
equal to one-half. So the denominator of our fraction
is equal to one-half cubed. Now, one-half cubed is
one-eighth. And one over one-eighth is simply
eight. So weβve evaluated d two π¦ by dπ₯
squared at π is equal to π by six to be equal to negative eight. Since negative eight is less than
zero, this tells us that, at π is equal to π by six, our curve is concave up.