Find the maximum value of $f(x) = x^3 - 12x^2 + 45x$ in the interval $[0, 7]$.

The sum of the maximum and minimum values of the function $f(x)=\frac{x^2-x+1}{x^2+x+1}$ is

The coordinates of the point $P$ on the graph of the function $y = e^{-|x|}$ where the portion of the tangent intercepted between the coordinate axes has the greatest area,is

Find two numbers whose sum is $24$ and whose product is as large as possible.

Let $x_0$ be the point of local minima of $f(x) = \overline{a} \cdot (\overline{b} \times \overline{c})$ where $\overline{a} = x \hat{i} - 2 \hat{j} + 3 \hat{k}$,$\overline{b} = -2 \hat{i} + x \hat{j} - \hat{k}$,and $\overline{c} = 7 \hat{i} - 2 \hat{j} + x \hat{k}$. Then the value of $\overline{a} \cdot \overline{b}$ at $x = x_0$ is:

The minimum value of $2x^2 + x - 1$ is