Differentiation

Step 1: Compute the first derivative :
Step 2: Factorize :
Step 3: Solve : These are the critical points where the slope of the tangent is zero.
Step 4: Compute the second derivative :
Step 5: Use the second derivative test to determine the nature of each critical point: At : At : At :
Step 6: Calculate the function values at critical points:
Step 7: Conclusion: Local maxima at ( ) and ( ). Local minima at ( ).

Step 1: The general equation of a circle touching both axes is: where is the radius of the circle, and the center is at .
Step 2: Differentiate implicitly with respect to :
Step 3: Eliminate using the relationship : Substitute into the equation:

Step 1: Rewrite the inequality: Step 2: The inequality represents the region above the line , excluding the points on the line.
Since , the origin is not part of the solution.

Step 1: The given differential equation is: Rewriting it, we get:
Step 2: The equation is of the first order as the highest derivative involved is .
Step 3: The equation does not involve any fractional power of the highest derivative, and the degree is the exponent of the highest derivative, which is 1.