Problem 1 - Analysis

Answer the following questions concerning the polynomial function $f(x) = x^3 - 3x^2 + px + q$.

1.

Give the condition of the existence of the local maximum and minimum, and the values for them.

2.

Let $x_1$ and $x_2$ be the value of $x$ for which the maximum and minimum occur, respectively. Calculate $\int_{x_1}^{x_2}f(x)dx$.

1.

$$f'(x) = 3x^2 - 6x +p$$

$\Delta = b^2 - 4ac = 12(3-p) > 0$ iff $p<3$. In this case, the polynomial has too roots given by:

$$x^+ = \frac{6 + 2\sqrt{3}\sqrt{3-p}}{6} = 1 + \frac{\sqrt{3}}{3}\... ...quad x^- = \frac{6 - 2\sqrt{3}\sqrt{3-p}}{6} = 1 - \frac{\sqrt{3}}{3}\sqrt{3-p}$$

which are the value of the local minimum and the local maximum respectively:

$$f(x) = x^3 - 3x^2 + px + q$$

The value of the minimum is:

$$f(x^+) = -2 + p + q + \sqrt{3}\sqrt{3-p}\left(-\frac{2}{3} + \frac{p}{3}\right)$$

$$f(x^+) = -2 + p + q + \sqrt{3}\sqrt{3-p}\left(\frac{2}{3} - \frac{2p}{9}\right)$$

2.

$$\int_{1-\frac{\sqrt{3}}{3}\sqrt{3-p}}^{1+\frac{\sqrt{3}}{3}\sqrt{... ...} + qx\right]_{1-\frac{\sqrt{3}}{3}\sqrt{3-p}}^{1+\frac{\sqrt{3}}{3}\sqrt{3-p}}$$