For $x + y = 8$,what is the maximum value of $xy$?

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 $(0, \infty)$

$A$ triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length $x$. The maximum area enclosed by the park is

Let $f(x)$ be a polynomial of degree four having extreme values at $x=1$ and $x=2$. If $\mathop {\lim }\limits_{x \to 0} \left[ {1 + \frac{{f(x)}}{{{x^2}}}} \right] = 3$,then $f(2)$ is equal to:

What is the minimum value of $y = 2x^3 - 21x^2 + 36x - 20$?

The sum of two numbers is fixed. Then their product is maximum when: