We are given the function: We need to find where this function has a local minimum. Step 1: Find the first derivative of . To find the critical points, we first take the derivative of : Using the power rule for derivatives: Step 2: Set the first derivative equal to zero to find critical points. To find the critical points, set : Solving for : Multiplying both sides by and simplifying: Thus, the critical points are and . Step 3: Determine the nature of the critical points using the second derivative. To determine whether these critical points correspond to local minima or maxima, we take the second derivative of : Using the power rule again: Now, evaluate at and : - At , , which is positive, indicating a local minimum at .
- At , , which is negative, indicating a local maximum at . Step 4: Conclusion. Since corresponds to a local minimum, the correct answer is option (A)