The maximum value of $x^{2/3} + (x-2)^{2/3}$ is

If $f(x) = \sqrt{x^2 + x} + \frac{\tan^2 \alpha}{\sqrt{x^2 + x}}$,where $\alpha \in (0, \pi/2)$ and $x > 0$,find the minimum value of $f(x)$.

Let $x=-1$ and $x=2$ be the critical points of the function $f(x)=x^3+ax^2+b \ln|x|+1, x \neq 0$. Let $m$ and $M$ respectively be the absolute minimum and the absolute maximum values of $f$ in the interval $\left[-2, -\frac{1}{2}\right]$. Then $|M+m|$ is equal to (Take $\ln 2 \approx 0.7$):

The number of real roots of the equation $e^{4x} + 2e^{3x} - e^{x} - 6 = 0$ is:

$P(x) = x^4 + ax^3 + bx^2 + cx + d$ is such that $x = 0$ is the only real root of $P'(x) = 0$. If $P(-1) < P(1)$,then in the interval $[-1, 1]$:

The minimum value of the polynomial $x(x + 1)(x + 2)(x + 3)$ is