Find the critical point of the function

\(\frac{{{{\sin }^{ - 1}}\left( {{y^2}} \right).\left( {{y^2} + 3y} \right).\left( {\sin \left( {{y^6} + 7y} \right)} \right)}}{{\left( {{y^9} + {y^{10}}} \right)}} + 10x\)

Concept:

1. Critical points of a function:

• For any function f(x) the critical points are defined as the points at which the derivative of the function is 0 or undefined.
• Usually, these points on further investigation represent maxima or minima of the function.

2. First derivative test:

• For any function, if the derivative changes its sign at a point then that point is a critical point of the function.
• If the first derivative is positive then the function is increasing and decreases if the derivative is negative.
• To find the critical point we equate the first derivative to 0.

3. Second derivative test:

• For any function, if the second derivative is positive at a critical point then that point indicates the minima.
• For any function, if the second derivative is negative at a critical point then that point indicates the maxima.
• For any function, if the second derivative is zero then the test is inconclusive and we can investigate further only by the nature of the curve.

Calculation:

Given:

\(f(x,~y)\) = \(\frac{{{{\sin }^{ - 1}}\left( {{y^2}} \right).\left( {{y^2} + 3y} \right).\left( {\sin \left( {{y^6} + 7y} \right)} \right)}}{{\left( {{y^9} + {y^{10}}} \right)}} + 10x~\)

First, differentiate the given with respect to x.

\(f_x~\) = 10 which is not equal to zero.

So, the critical point of the function does not exists.