Video Transcript
Suppose π prime of four equals
zero and π double prime of four equals negative four. What can you say about π at the
point π₯ equals four? π has a local minimum at π₯ equals
four. π has a local maximum at π₯ equals
four. π has a point of inflection at π₯
equals four. It is not possible to state the
nature of the turning point of π at π₯ equals four. Or π has a vertical tangent at π₯
equals four.
Letβs take each of the pieces of
information weβve been given in turn. Firstly, weβre told that π 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 π has a critical
point when π₯ is equal to four. Next, weβre told that π double
prime of four is equal to negative four. So the second derivative of our
function π is negative when π₯ is equal to four. The second derivative will be
negative at a local maximum. So we can conclude that π has a
local maximum at π₯ equals four.
Thatβs the second option in the
list weβve been given. The first, third, and fourth
options are therefore false. If a point is a local maximum, it
canβt 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. Letβs consider the fifth
option. We know that the first derivative
of our function π is zero when π₯ is equal to four, which means that the slope of
the curve and the slope of the tangent will be zero. Therefore, π will have a
horizontal, not a vertical tangent at π₯ equals four. So weβve completed the problem. π has a local maximum at π₯ equals
four.