For the function $\displaystyle x^3-x^2+4x-3$, find:

1. Intervals of increase and decrease

2. the location of any maximum or minimum points

3. the intervals of concavity up or down

4. The locations of any points of inflection.

The first thing I do is differentiate for the first and second derivatives:

$\displaystyle f'(x) = 2x^2-2x+4$

$\displaystyle f''(x) = 6x-2$

My attempts:

1. Since the derivative of the function has no zeros, the function is continuously increasing for all values of x and has no interval of decrease.

2. There aren't any local maximums or minimums, since the function is continuously increasing and has no negative slopes.

3. I set the second derivative to zero and solve for x, which gives me a value of $\displaystyle x = \frac{1}{3}$ so when $\displaystyle f(x) < 0, x < \frac{1}{3} $ which means the function is concave down over this interval, conversely: $\displaystyle f(x) > 0, x > \frac{1}{3}$ which would mean the function is concave up over this interval.

4. The point of inflection is $\displaystyle x = \frac{1}{3}$

Can anyone clarify to me that my answers are correct? I haven't seen an example like this in my notes. I appreciate any reply and thanks in advance.