What is the minimum value of the sum of a real number $x$ and its reciprocal?

Maximum value of the function $f(x) = [x(x-1) + 1]^{\frac{1}{3}}$ for $0 \leq x \leq 1$ is . . . . . . .

Find the local maximum and local minimum values for the function $f(x) = x\sqrt{1 - x}$ where $0 < x < 1$.

Let $S$ be the set of all twice differentiable functions $f$ from $R$ to $R$ such that $\frac{d^2 f}{d x^2}(x) > 0$ for all $x \in (-1, 1)$. For $f \in S$,let $X_f$ be the number of points $x \in (-1, 1)$ for which $f(x) = x$. Then which of the following statements is(are) true?
$(A)$ There exists a function $f \in S$ such that $X_f = 0$
$(B)$ For every function $f \in S$,we have $X_f \leq 2$
$(C)$ There exists a function $f \in S$ such that $X_f = 2$
$(D)$ There does $NOT$ exist any function $f$ in $S$ such that $X_f = 1$

Find two positive numbers whose sum is $16$ and the sum of whose cubes is minimum.

The difference between the absolute maximum and absolute minimum values of the function $f(x)=2x^3-15x^2+36x-30$ on the interval $[-1, 4]$ is: