### Video Transcript

Determine the inflection points of
the curve ๐ฆ equals ๐ฅ squared plus two ๐ฅ minus five.

First of all, letโs remind
ourselves what an inflection point is. Itโs a point where the concavity
changes. We go from concave up to concave
down or concave down to concave up. To identify inflection points, we
use the second derivative. So, letโs visualize why the second
derivative helps us to find inflection points.

If this is a graph of a function
๐, not necessarily our function, we can see that this part of the graph is concave
down and this part is concave up. So, the point in-between is an
inflection point. Where the graph is concave down,
the slope is decreasing. And where the graph is concave up,
the slope is increasing. So, if we graph the derivative of
this function ๐ prime, which is the slope function, it would look like this. Decreasing, while the graph of ๐
is concave down. And after the point of inflection,
itโs increasing where the graph of ๐ is concave up.

What about the second
derivative? Well, the slope of the graph of ๐
prime of ๐ฅ is negative before the point of inflection and itโs positive after. So, what does this tell us? A point of inflection occurs where
the second derivative changes sign from negative to positive or positive to
negative. So, thatโs why the second
derivative helps us to find inflection points. Letโs go ahead and find our second
derivative.

To do this, letโs remember the
general power rule that the derivative of ๐๐ฅ to the power of ๐ is ๐๐๐ฅ to the
power of ๐ minus one. So, for our function of ๐ฅ, ๐ฅ
squared plus two ๐ฅ minus five, ๐ prime of ๐ฅ equals two ๐ฅ plus two. Thatโs because ๐ฅ on its own is the
same as ๐ฅ to the power of one. So, two ๐ฅ differentiates to
two. And constants like negative five
differentiate to zero. And now, we differentiate again to
get our second derivative. This gives us ๐ double prime of ๐ฅ
equals two.

Remember, we said that, to find
inflection points, weโre looking for points where the second derivative function
changes sign. But we got a positive constant, so
what does this mean? Well, ๐ of ๐ฅ is concave up where
the second derivative is positive. As the second derivative is a
constant, it canโt change sign. So, the graph of ๐ฆ must always be
concave up. So, as the concavity doesnโt
change, we conclude that the curve has no inflection points. In fact, if we draw a sketch of
this function, we can see that the curve is always concave up. There are no points of inflection
on this curve.