Let $$f\left( {x,y} \right) = {{a{x^2} + b{y^2}} \over {xy}}$$, where $$a$$ and $$b$$ are constants. If $${{\partial f} \over {\partial x}} = {{\partial f} \over {\partial y}}$$ at x = 1 and y = 2, then the relation between $$a$$ and $$b$$ is

Taylor series expansion of $$f\left( x \right) = \int\limits_0^x {{e^{ - \left( {{{{t^2}} \over 2}} \right)}}} dt$$ around 𝑥 = 0 has the form

f(x) = $${a_0} + {a_1}x + {a_2}{x^2} + ...$$

The coefficient $${a_2}$$ (correct to two decimal places) is equal to _______.

The integral $$\int\limits_0^1 {{{dx} \over {\sqrt {\left( {1 - x} \right)} }}} $$ is equal ________.

As $$x$$ varies from $$- 1$$ to $$3,$$ which of the following describes the behavior of the function $$f\left( x \right) = {x^3} - 3{x^2} + 1?$$