Let $a$ be a real number such that the function $f(x) = ax^2 + 6x - 15, x \in R$ is increasing in $(-\infty, \frac{3}{4})$ and decreasing in $(\frac{3}{4}, \infty)$. Then the function $g(x) = ax^2 - 6x + 15, x \in R$ has a:

The local maximum value $l$ and local minimum value $m$ of $f(x) = \frac{x^2+2x+2}{x+1}$ in $R - \{-1\}$ exist at $\alpha, \beta$ respectively, then $\frac{l+m}{\alpha+\beta} =$

$A$ tank with a rectangular base and rectangular sides,open at the top is to be constructed so that its depth is $4 \ m$ and volume is $36 \ m^3$. If building of the tank costs $₹ 100$ per square meter for the base and $₹ 50$ per square meter for the sides,then the cost of the least expensive tank is:

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 $n$

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

If $x$ is real,then the greatest and least values of $\frac{x^2 - x + 1}{x^2 + x + 1}$ are