The maximum value of $f(x) = (x + 1)^{\frac{1}{3}} - (x - 1)^{\frac{1}{3}}$ for $x \in [0, 1]$ is ....

Given $f(x) = -\frac{x^3}{3} + x^2 \sin(1.5a) - x \sin(a) \sin(2a) - 5 \sin^{-1}(a^2 - 8a + 17)$,then:

Let $\exp(x)$ denote the exponential function $e^x$. If $f(x) = \exp\left(x^{\frac{1}{x}}\right)$ for $x > 0$, then the minimum value of $f$ in the interval $[2, 5]$ is:

Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that of the cone.

The cost function at American Gadget is $C(x) = x^3 - 6x^2 + 15x$ (where $x$ is in thousands of units and $x > 0$). The production level at which the average cost is minimum is:

Local maximum and local minimum values respectively of the function $f(x)=(x-1)(x+2)^2$ are