1. ## Differentiation

f(x)=x^3-3x^2+3x
f '(x)=3x^2-6x+3
I have no issue with finding f ' (x).

The next part of the question says, thus show that f' (x) => 0 for all x and interpret this in terms of the graph f(x)=x^3-3x^2+3x

If anyone can help that would be great.

2. Originally Posted by thekiterunner
f(x)=x^3-3x^2+3x
f '(x)=3x^2-6x+3
I have no issue with finding f ' (x).

The next part of the question says, thus show that f' (x) => 0 for all x and interpret this in terms of the graph f(x)=x^3-3x^2+3x

If anyone can help that would be great.
Simply solve the inequality $3x^2-6x+3 \geq 0$ by using the fact that $3x^2-6x+3 = 3(x^2-2x+1)$.

After you proved that, you know $f'(x) \geq 0 \ \forall x \in \mathbb{R}$. What does this tell you about the $f(x)$? What, in general, does $f'(x)$ mean in regards to $f(x)$?

3. i do not undertsand that one little bit...any chance for simplier terms

4. If you have a general quadratic equation:
$f(x)=ax^2+bx+c$

And let the discriminant $D=b^2-4\cdot ac$.

Now if D<0 that means the equation has no real solution, that is the graph nowhere goes
through the x-axis. It is entirely contained on either side. Either $f<0$ or $f>0$ for all x.
Now if you dont know if f is below or above the x-axis,
just check for some point if it is above or below, then you know that the rest of the points
are on the same side.

If D=0 then there is one real solution, the graph is entirely on one side and "touches" the
x-axis in only one point, it never crosses the axis. So either $f\leq0$ or $f\geq0$ for all x.

What if D>0 ?

But anyway what is D in your equation?

After you have found out the first part, just think about, like Defunkt said, what $f'(x)$ means in regard to $f(x)$.

Does that help?

5. But anyway what is D in your equation?

D=6^2 - (4 x 3 x 3) = 0

So becuase D=0 i have one real solution

f'(x) is just the gradient function of f(x)

6. So if f'(x) is always positive or zero, then f(x) must be increasing (not strictly).
And f(x) does not have a local maxima or minima at it's critical point.

7. The geometrical meaning of f'(x) regarding f(x) is the slope -- f'(x0) is (if it exists) the slope of the function f in point x0.

Now, if $\forall x \in \mathbb{R}, \ f'(x) \geq 0$, what does this tell you about the graph of f?