i have a calculus question about a question? help please!?

Question

OK so on this is the question:
if f(x)= x^4 - 4x^3 - 36x^2 + 9
find the critical values, make the sign diagram of f(x) (prime) and f(x) (double prime)....

so when i work it out for the critical values i factor the derivative equation and get....
4x(x+3)(x-6)=0
and i know u set (x+3) and (x-6) equal to 0 which you get x=-3 and x=6 but I want to know is,
do you set the 4x equal to zero too? and if you do is it also a critical value?

thanks for helping

Answer 1

Yes. Set the 4x equal to zero too. It is a critical value, either a max, a min, or an inflection point.

Answer 2

Yes, you do. And yes, it is. All three of the roots (including x = 0) of the derivative equate to x values for which the tangent line to f(x) is horizontal and represent points where f(x) is passing through a local maximum or minimum. The sign of the 2'nd derivative at those x values will tell you if the point is a maximum (the 2'nd derivative will be negative) or a minimum (the 2'nd derivative will be positive). In the case of the f(x) you have, the curve will 'kinda' resemble a large 'W' if you graph it out. The minimums will be at x = -3 and x = 6, and the maximum will be at x = 0. (And don't worry, after 40+ years in math, you too will be able to look at an equation and see what its 'shape' will look like ☺)

HTH

Doug

Answer 3

you also have to consider the 4x since it is a factor.

0 is a critical number.

4x=0 therefore x=0 and if x is zero the equailty holds. and the Leftside equals the right.