Video Transcript
Suppose 𝑓 prime of four equals
zero and 𝑓 double prime of four equals negative four. What can you say about 𝑓 at the
point 𝑥 equals four? 𝑓 has a local minimum at 𝑥 equals
four. 𝑓 has a local maximum at 𝑥 equals
four. 𝑓 has a point of inflection at 𝑥
equals four. It is not possible to state the
nature of the turning point of 𝑓 at 𝑥 equals four. Or 𝑓 has a vertical tangent at 𝑥
equals four.
Let’s take each of the pieces of
information we’ve been given in turn. Firstly, we’re told that 𝑓 prime
of four is equal to zero. And if the first derivative of a
function is equal to zero at a given point, then the function has a critical point
at that point. So we know that 𝑓 has a critical
point when 𝑥 is equal to four. Next, we’re told that 𝑓 double
prime of four is equal to negative four. So the second derivative of our
function 𝑓 is negative when 𝑥 is equal to four. The second derivative will be
negative at a local maximum. So we can conclude that 𝑓 has a
local maximum at 𝑥 equals four.
That’s the second option in the
list we’ve been given. The first, third, and fourth
options are therefore false. If a point is a local maximum, it
can’t also be a local minimum or a point of inflection. And it has been possible for us to
determine the nature of this turning point. Let’s consider the fifth
option. We know that the first derivative
of our function 𝑓 is zero when 𝑥 is equal to four, which means that the slope of
the curve and the slope of the tangent will be zero. Therefore, 𝑓 will have a
horizontal, not a vertical tangent at 𝑥 equals four. So we’ve completed the problem. 𝑓 has a local maximum at 𝑥 equals
four.
