The greatest value of the real-valued function $f(x)=(x+1)^{1/3}-(x-1)^{1/3}$ on the interval $[0, 1]$ is:

If $f(x)=3x^3-9x^2-27x+15$,then the maximum value of $f(x)$ is $.....$

$A$ wire of length $28 \, m$ is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?

Let the function $f: (0, \pi) \rightarrow R$ be defined by $f(\theta) = (\sin \theta + \cos \theta)^2 + (\sin \theta - \cos \theta)^4$. Suppose the function $f$ has a local minimum at $\theta$ precisely when $\theta \in \{\lambda_1 \pi, \dots, \lambda_r \pi\}$,where $0 < \lambda_1 < \dots < \lambda_r < 1$. Then the value of $\lambda_1 + \dots + \lambda_r$ is:

Find the maximum value of $(1/x)^x$.

The function $f(x) = x^{25}(1 - x)^{75}$ for $x \in [0, 1]$ attains its maximum at $x = \dots$