Finding intervals of increase and decrease from an equation of f(x)

[pbonnie]

Homework Statement

For the function $f(x) = x^3 - x^2 + 4x - 3$

Homework Equations

The Attempt at a Solution

I found the first and second derivative, the first derivative is $f'(x) = 3x^2 - 2x + 4$
This is not factorable, as the discriminant is < 0, meaning there is no x-intercepts.
Does this mean that because f'(x) is positive for all values of x, f(x) is increasing for all values of x? And if not, I'm not sure what the next step is.
Thank you for you help

 

[CompuChip]

Have you sketched the graph of the function, e.g. using your graphic calculator or computer software (like Google or Wolfram Alpha)?

Does the function look everywhere increasing? That would give you confidence that your calculations are correct and your conclusion is justified.

 

[pbonnie]

I did type in the f(x) function, which has intervals of increase AND decrease.. which is why I'm not very confident in my answer. But now I'm stuck as to how to figure out the correct answer. I'm doing an online course and it hasn't explained this part.

 

[Mark44]

I did type in the f(x) function, which has intervals of increase AND decrease.. which is why I'm not very confident in my answer.

I think you might have entered the formula incorrectly. For the function you show, f'(x) > 0, for all real x, so f is increasing everywhere.

But now I'm stuck as to how to figure out the correct answer. I'm doing an online course and it hasn't explained this part.

 

[pbonnie]

Oh okay, I tried it on a different graphing program and it worked. Thank you both!
For a part b of the question, it asks for the location of any maximum or minimums. Because it is constantly increasing, this means there are no maximum or minimum values?

 

[Mark44]

There are no points at which the derivative is zero, or at which the function is undefined, or endpoints of a domain, so yes, there are no maximum or minimum points.