how to find turning points of a quartic function using calculus?

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There are at most three turning points for a quartic, and always at least one.

At a turning point (of a differentiable function) the derivative is zero. However the derivative can be zero without there being a turning point. (Consider $$f(x)=x^3$$ or $$f(x)=x^5$$ at $$x=0$$).

A good strategy for kinds of functions you don't completely understand is to sketch them - this works well for polynomials. What is the general shape of a quartic with positive coefficient of $$x^4$$?

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Nyx Smith

Updated on August 01, 2022

• Nyx Smith 3 months

My teacher gave a function of $$f(x)= 2x^4-3x^3-21x^2+16x+60$$ He said that there needs to be $$4$$ turning points. I only now how to find the turning points if the function is at cubic not quartic.

• A. Goodier about 3 years
Did you try differentiating and setting the derivative $=0$?
• Nyx Smith about 3 years
I'm not sure. This topic is new so I really have no idea what to do
• A. Goodier about 3 years
Do you know how to work out $f'(x)$?
• Nyx Smith about 3 years
No. I'm pretty hopeless at this. I'm not really good with math or other terms related to it.
• A. Goodier about 3 years
To be honest, it is difficult to help you with this if you do not know how to differentiate. I suggest you practice differentiating simple functions on Khan Academy or similar before attempting this question.
• Toby Mak about 3 years
What is your definition of 'turning point'? Many people have interpreted it as a point of 'inflection point' where $f''(x) = 0$, but you can also interpret it as where the function 'turns' or where $f'(x) = 0$.
• Nyx Smith about 3 years
what method do i use in order to get 3 x-coordinates?
• Toby Mak about 3 years
You have that $x=2$ by the rational root theorem. Dividing $f'(x)$ by $x-2$ gives the other two roots, since the other factor is a quadratic. You should use the quadratic formula to do this, since you will find the quadratic cannot be factored.
• Angina Seng about 3 years
I get $f'(x)=8x^3-9x^2-42x+16$.
• Toby Mak about 3 years
That was a silly mistake, thanks. The rest of the post should not be affected.
• Angina Seng about 3 years
I don't get $f'(2)=0$.