### Video Transcript

Determine the local maximum and
minimum values of the function 𝑦 equals negative three 𝑥 squared minus six 𝑥
minus four.

First, we recall that at critical
points, the first derivative of the function — in this case d𝑦 by d𝑥 — is equal to
zero. So our first step is going to be to
find the first derivative of this function. By applying the power rule of
differentiation, we find that d𝑦 by d𝑥 is equal to negative six 𝑥 minus six. We then set our expression for d𝑦
by d𝑥 equal to zero and solve for 𝑥, giving 𝑥 is equal to negative one. Our function, therefore, has one
critical point, which occurs when 𝑥 is equal to negative one.

Next, we need to evaluate the
function at the critical point, which we do by substituting 𝑥 equals negative one
into the equation we’ve been given. We obtain 𝑦 equals negative three
multiplied by negative one squared minus six multiplied by negative one minus four
which simplifies to negative one. This tells us then that the only
critical point of this function is the point with coordinates negative one, negative
one. But we need to determine whether
this is a local minimum or local maximum, which we’ll do by applying the second
derivative test.

To find the second derivative, we
need to differentiate our first derivative with respect to 𝑥. So we’re finding the derivative of
negative six 𝑥 minus six with respect to 𝑥. Applying the power rule, we see
that this derivative is just equal to negative six. Now, this second derivative is
actually just a constant because we’ve differentiated a quadratic expression
twice. So we don’t need to substitute the
𝑥-coordinate at our critical point in in order to evaluate because the second
derivative is constant for all values of 𝑥. We note that negative six is less
than zero. We recall that if the second
derivative of a function is negative at the critical point, then the critical point
is a local maximum. So the point negative one, negative
one is indeed a local maximum of this function.

So we can conclude that this
function has no local minimum value but has a local maximum value of negative
one. Notice that is the value of the
function itself that we are giving here, not the 𝑥-value, although they are both
the same in this instance. We can also confirm this result
using our knowledge of the graphs of quadratic functions. As the coefficient of 𝑥 squared in
this curve is negative, the graph of this quadratic will be a negative parabola. We know that parabolas have only a
single critical point. And if the coefficient of 𝑥
squared is negative, then their critical point will be a local maximum.