Derivatives

Slope of tangent from origin is . For curve , equation of tangent:

For the function , the condition for a relative minimum at is that the derivative of the function at that point should be zero.
This implies , and can take any value.
Therefore, the correct answer is (2).

The function is analyzed by checking whether it is one-to-one and onto.
After analyzing the function, it is found to be onto but not one-to-one.

We are given the equation: To find , we differentiate both sides with respect to using implicit differentiation. 1. Differentiating the left-hand side: 2. Differentiating the right-hand side using the product rule: Now, equating both sides: 3. Solving for : Factor out : Thus, we get: Therefore, the correct answer is option (C) .