Differential Equations Answers

How to find maxima and minima. Process
1. to find x where dy/dx=o. let the x =a
2. then find d2y/dx2
3.if d2y/dx2 at x=a >0 then we will find minima for x=a
4.if d2y/dx2 at x=a <0 then we will find maxima for x=a
5.if d2y/dx2 at x=a =0 then we will have to proceed to the d3y/dx3
6.if d3y/dx3 at x=a in not eqal 0 , we will not find any maxima or minima.
7.if d3y/dx3 at x=a equals 0 then we will proceed to d4y/dx4. and if d4y/dx4 at x=a >0 then it's minima and if < 0 then maxima.

Please explain the rule 6 above. Why will we not find any maxima or minima in that case? Please explain with figure and example.

sin^-1 (-x)=??

sec^-1 (-x)=??

tan^-1(-x)=??

Cot^-1 (-x)=??

Cosec^-1 (-x)=??

explain in each case .

What is the differentiation of sec^-1 ((x^2+1)/(x^2-1))?

Why do we use x=0 to find f(0) in maclaurin series? why another numbers are not used?
and why f(0) is needed?

Find the solution of the Riccati equation:
dy/dx = 2cos^2 x - sin^2 x + y^2/ 2cosx ; y1(x) = sinx.

Find the value of b for which the equation
(y e^2xy + x) dx + bx e^2xy dy = 0
is exact, and hence solve it for that value of b .

Find the differential equation of the family of curves y = e^x (Acos x + Bsin x) where A and B are arbitrary constants and hence solve the equation.

Why maxima is found when second derivative is less than zero?
Why minima is found when second derivative is grater than zero?
Why and when the minima is grater than maxima. Please explain with example.

sketch the curve x^3-x^2-6x

if x-2y=6, find the greatest value of y