# When are we expected to prove a maximum? (1 Viewer)

#### ChrisChrisau

##### Member
Hey all,

If a question asks to find a maximum on a complex curve and there is only one turning point, are we still expected to find the double differential?

For example, in the 2005 Maths paper, question 10 (b) (iv)
http://www.boardofstudies.nsw.edu.au/hsc_exams/hsc2005exams/pdf_doc/maths_05.pdf

There is only one turning point here. Do we still have to double differentiate to prove that it is a maximum?

All insight is much appreciated Thanks!
Chris

#### imoO

##### Member
Hey all,

If a question asks to find a maximum on a complex curve and there is only one turning point, are we still expected to find the double differential?

For example, in the 2005 Maths paper, question 10 (b) (iv)
http://www.boardofstudies.nsw.edu.au/hsc_exams/hsc2005exams/pdf_doc/maths_05.pdf

There is only one turning point here. Do we still have to double differentiate to prove that it is a maximum?

All insight is much appreciated Thanks!
Chris
From personal experience, better safe than sorry. I think that solves your question (Yes, you should double differentiate to prove its max, or u can use the other method where you sub values into the first derivative)

#### scumm

##### New Member
finding the second derivative isn't absolutely necessary to prove that a turn pt is max/min. if you have the first derivative, input an x value that is on the left side of the point, and one that is on the right side. this'll give you the gradient on either side.

If the gradient on the left side is negative, and the right side is positive, then it's a min. turn. pt. If vica versa, then max turn point.

#### Albertingu

##### Member
finding the second derivative isn't absolutely necessary to prove that a turn pt is max/min. if you have the first derivative, input an x value that is on the left side of the point, and one that is on the right side. this'll give you the gradient on either side.

If the gradient on the left side is negative, and the right side is positive, then it's a min. turn. pt. If vica versa, then max turn point.
Yeah this.

Though most of the time it is just as easy to find the second derivative.

#### tonyharrison

##### Member
If y'' < 0 , then maximum turning point...you don't need to prove, only if it's a possible point of inflexion (y'' = 0)

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#### annabackwards

##### <3 Prophet 9
ALWAYS PROVE IT IS A MIN/MAX TURNING POINT.

Either double differentiate if it's easy and prove that it's >0 or <0 at the turning pt to prove it's a min or max.

Otherwise draw up a table and show that there's a sign change on either side as suggested before.

You will lose at least a mark (if it is out of 2) or lose the entire mark if it is out of 1 if you don't find it's nature.

#### Trebla

If the question just says for example:

"Find the turning points of y = f(x)"
OR
"Show that (0,0) is a turning point of y = f(x)"

You can get away without proving the nature of the turning point you found, because it never specified whether they should be minimum or maximum. They just want the coordinates of the turning points full stop.

If the question says for example:

"Find the turning points of y = f(x) and determine their nature"
OR
"Show that there is a minimum turning point at the origin for y = f(x)"

You MUST determine (either by second derivative or by investigating the neighbourhood of the turning point) whether the turning point you found is maximum or minimum.

#### lychnobity

##### Active Member
Otherwise draw up a table and show that there's a sign change on either side as suggested before.

You will lose at least a mark (if it is out of 2) or lose the entire mark if it is out of 1 if you don't find it's nature.
To elaborate on this point, I'd suggest actually substituting values into the calculator.

#### MOP777

##### New Member
Even if it says "find the maximum turning point" and theres only one, check it anyway, the max could still be at one of the endpoints, but I'm pretty sure they wouldn't do that, you should still check it anyway.

#### anom1ly

##### Member
In my trial, we were asked to find the turning points of a curve, then find the point of inflexion. We all lost marks because noone tested the point of inflexion. Is this common?

#### Bdogz

##### Member
In my trial, we were asked to find the turning points of a curve, then find the point of inflexion. We all lost marks because noone tested the point of inflexion. Is this common?
Yes you even have to prove that there is a change in concavity on either side of the point of inflexion.

#### anom1ly

##### Member
Yes you even have to prove that there is a change in concavity on either side of the point of inflexion.
will definately keep in mind for tomorrow. thanks.

#### ChrisChrisau

##### Member
Thanks for the replies everyone Will make sure I do that for tomorrow!