$A$ box open from the top is made from a rectangular sheet of dimensions $a \times b$ by cutting squares of side $x$ from each of the four corners and folding up the flaps. If the volume of the box is maximum,then $x$ is equal to:

For every twice differentiable function $f : R \rightarrow [-2, 2]$ with $(f(0))^2 + (f'(0))^2 = 85$,which of the following statement$(s)$ is (are) $TRUE$?
$(A)$ There exist $r, s \in R$,where $r < s$,such that $f$ is one-one on the open interval $(r, s)$.
$(B)$ There exists $x_0 \in (-4, 0)$ such that $|f'(x_0)| \leq 1$.
$(C)$ $\lim_{x \rightarrow \infty} f(x) = 1$.
$(D)$ There exists $a \in (-4, 4)$ such that $f(a) + f''(a) = 0$ and $f'(a) \neq 0$.

Let $f: R \rightarrow R$ be given by $f(x)=(x-1)(x-2)(x-5)$. Define $F(x)=\int_0^x f(t) dt$ for $x>0$. Which of the following options is/are correct?
$(1)$ $F$ has a local minimum at $x=1$
$(2)$ $F$ has a local maximum at $x=2$
$(3)$ $F(x) \neq 0$ for all $x \in (0,5)$
$(4)$ $F$ has two local maxima and one local minimum in $(0, \infty)$

The local maximum of $y=x^3-3 x^2+5$ is attained at

Let $f(x) = \frac{\sin \pi x}{x^2}, x > 0$. Let $x_1 < x_2 < x_3 < \ldots < x_n < \ldots$ be all the points of local maximum of $f(x)$ and $y_1 < y_2 < y_3 < \ldots < y_n < \ldots$ be all the points of local minimum of $f(x)$. Which of the following statements are true?
$(1)$ $|x_n - y_n| > 1$ for every $n$
$(2)$ $x_1 < y_1$
$(3)$ $x_n \in (2n, 2n + \frac{1}{2})$ for every $n$
$(4)$ $x_{n+1} - x_n > 2$ for every $n$

If $x$ is real,then the minimum value of $\frac{x^2-x+1}{x^2+x+1}$ is