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C3 - Difficult stationary point

10

I'm sure it is something simple I'm missing, but any help would be appreciated

OCR - January 2008, Q7 ii)

Given the curve has exactly one stationary point, find the value of k, and determine the exact co-ordinates of the stationary point

The equation of the line, already differentiated =

e^2x (2x^2 + 2kx + k)
__________________
(x + k)^2

So far,

e^2x (2x^2 + 2kx + k)
__________________ = 0
(x + k)^2

e^2x (2x^2 + 2kx + k) = 0

(2x^2 + 2kx + k) = 0

Now where do I go? And no, there were no points given, or x values or anything to go on :confused:

Reply 1

DFranklin

18

If there's exactly one stationary point, how many roots can (2x^2+2kx+k) have? What does that tell us about "B^2-4AC"?

Reply 2

Merlinedge

6

As the curve has exactly one stationary point the solution to the equation must have 1 repeated root.

If you remember back to C1 or C2 (cant remember which) - it means the discriminant of the quadratic is equal to 0.

That should give you k.

Reply 3

__Katy

OP

10

Of course! The discriminant! I knew it was something simple, thank you :p:

Reply 4

TMCkins

Original post

by Merlinedge

As the curve has exactly one stationary point the solution to the equation must have 1 repeated root.

If you remember back to C1 or C2 (cant remember which) - it means the discriminant of the quadratic is equal to 0.

That should give you k.

I know this is abit later lol, but i thought the discriminant gave you the number of roots, the question just tells you that there is one stationary point, how can we know that this stationary point is also a root?

Reply 5

TMCkins

Original post

by DFranklin

If there's exactly one stationary point, how many roots can (2x^2+2kx+k) have? What does that tell us about "B^2-4AC"?

I know this is abit later lol, but i thought the discriminant gave you the number of roots, the question just tells you that there is one stationary point, how can we know that this stationary point is also a root?

Reply 6

davros

16

Original post

by TMCkins

I know this is abit later lol, but i thought the discriminant gave you the number of roots, the question just tells you that there is one stationary point, how can we know that this stationary point is also a root?

A stationary point is precisely one where the first derivative is zero. Therefore the discriminant tells you how many stationary points there can be, since in this case they arise as the zeros of a quadratic expression.

Reply 7

Branny101

Original post

by davros

A stationary point is precisely one where the first derivative is zero. Therefore the discriminant tells you how many stationary points there can be, since in this case they arise as the zeros of a quadratic expression.

When x values are found why is x=1 negated? As I found both x=1 and -1?!

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Reply 8

DFranklin

18

If you're talking about the same question as the original post, you're told there's exactly 1 stationary point. If you found x=1 and x=-1, you made a mistake somewhere.

Reply 9

Branny101

Original post

by DFranklin

If you're talking about the same question as the original post, you're told there's exactly 1 stationary point. If you found x=1 and x=-1, you made a mistake somewhere.

Yeah I was, sorry to butt-in :frown:

Okay thanks :smile:

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(edited 11 years ago)

C3 - Difficult stationary point

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