Calculating turning points

[Anthonyk2013]

calculate the turning points and max and min value on the curve Y=x^3-3x^3-9x+10.

started a course and its been a few years sense I have done these questions.

How do I factorise Y=x³-3x²-9x+10.

 

[HallsofIvy]

calculate the turning points and max and min value on the curve Y=x^3-3x^3-9x+10.

started a course and its been a few years sense I have done these questions.

How do I factorise Y=x³-3x²-9x+10.

To answer your last question, I presume you know that x- a is a factor of polynomial P(x) if and only if x= a satisfies P(x)= 0. Looking for factors is the same as looking for roots to the equation. By the "rational root theorem" any rational number satisfying the equation must be an integer factor of 10- that is it must be one of 1, -1, 2, -2, 5, -5, 10, or -10. But it is easy to see, just by putting those numbers into the polynomial, that none of those satisfy the equation. So there is NO rational a such that x- a satisfies this equation. There is a "cubic formula" you could apply to find non-rational roots but that is very complicated.

However, why do you want to solve that equation? The question does not ask where y is equal to 0 (I am assuming you meant y= x^3- 3x^2- 9x+ 10) but where it as turning points. Those occur were the derivative of y is 0. And the derivative is a quadratic.

 

[Anthonyk2013]

I attached what I think is the solution.

 

[Ishuda]

I attached what I think is the solution.

Yes, you have found the turning points and the local min (x=3) and max (x=-1). The curve has no overall (global) min and max as it goes to $\displaystyle \pm\infty$ as x goes to $\displaystyle \pm\infty$, respectively.

 

[Steven G]

If you have a cubic y = ax^3 + bx^2 + cx =d then the relative max/min occur at x=[-b+/-sq rt (b^2-3ac)]/3a (if there is a max and min)and the point of inflection (turning point) occurs at x=-b/3a.
Can you see why these equations work??
Jomo

 

[Anthonyk2013]

If you have a cubic y = ax^3 + bx^2 + cx =d then the relative max/min occur at x=[-b+/-sq rt (b^2-3ac)]/3a (if there is a max and min)and the point of inflection (turning point) occurs at x=-b/3a.
Can you see why these equations work??
Jomo

Think so, only my second time doing one so practice make perfect.