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# Quick maths help! Cubic turning points. watch

1. Quick question about the number of turning points on a cubic - I'm sure I've read something along these lines but can't find anything that confirms it!

If a cubic has two turning points, then the discriminant of the first derivative is greater than 0.
If it has one turning point (how is this possible?) then the discriminant of the derivative = 0.
And if there are no turning points, the discriminant of the derivative is less than 0.

If you know whether this is true or not, please let me know!
2. (Original post by ksma1999)
Quick question about the number of turning points on a cubic - I'm sure I've read something along these lines but can't find anything that confirms it!

If a cubic has two turning points, then the discriminant of the first derivative is greater than 0.
If it has one turning point (how is this possible?) then the discriminant of the derivative = 0.
And if there are no turning points, the discriminant of the derivative is less than 0.

If you know whether this is true or not, please let me know!
Yes this is true. For the second one, consider where there is only 1 turning point - the point of inflection.

These can easily be verified because when you differentiate a cubic you end up with a quadratic. You make that quadratic equal to 0 and it's a simple case of considering whether there are 2 roots, only 1 root or no roots whatsoever hence the use of discriminant.
3. (Original post by ksma1999)
Quick question about the number of turning points on a cubic - I'm sure I've read something along these lines but can't find anything that confirms it!

If a cubic has two turning points, then the discriminant of the first derivative is greater than 0.
If it has one turning point (how is this possible?) then the discriminant of the derivative = 0.
And if there are no turning points, the discriminant of the derivative is less than 0.

If you know whether this is true or not, please let me know!
Turning points occur when the derivative is zero. A cubic will have a quadratic derivative. If the discriminant is:
- Greater than zero => two real solutions, so two turning points
- Zero => repeated root, so one turning point
- Less than zero => no real solutions, so no turning points
4. (Original post by RDKGames)
Yes this is true. For the second one, consider where there is only 1 turning point - the point of inflection.

These can easily be verified because when you differentiate a cubic you end up with a quadratic. You make that quadratic equal to 0 and it's a simple case of considering whether there are 2 roots, only 1 root or no roots whatsoever hence the use of discriminant.

Ah yes thank you, that makes sense! I wanted to actually understand it rather than just blindly following a method.
5. (Original post by RogerOxon)
Turning points occur when the derivative is zero. A cubic will have a quadratic derivative. If the discriminant is:
- Greater than zero => two real solutions, so two turning points
- Zero => repeated root, so one turning point
- Less than zero => no real solutions, so no turning points
Great, thank you for your help! I get it now!
6. (Original post by ksma1999)
Quick question about the number of turning points on a cubic - I'm sure I've read something along these lines but can't find anything that confirms it!

If a cubic has two turning points, then the discriminant of the first derivative is greater than 0.
If it has one turning point (how is this possible?) then the discriminant of the derivative = 0.
And if there are no turning points, the discriminant of the derivative is less than 0.

If you know whether this is true or not, please let me know!
A cubic can not have only one turning point but it can have only one stationary point.
If you think about these statements then they do make sense. If a cubic has two turning points then the derivative of the equation of the curve must have 2 roots (real) - if it didn't there wouldn't be 2 turning points would there?

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