Calculus

Rolle’s theorem guarantees at least one root if the function is continuous and differentiable, and the endpoints have the same value.

Step 1: To find the angle between two curves, we first find the gradients of the tangent lines to each curve at the point of intersection.
Step 2: Use the formula for the angle between two curves: where and are the slopes of the tangents to the curves. The angle is found to be .

Final Answer:

Step 1: To find the interval where the function is decreasing, we need to compute the derivative of the function.
Step 2: The first derivative is .
Step 3: Solve , and we find the critical points. The function is decreasing on .

Final Answer:

Step 1: Understanding the Given Determinants
Given that and using the determinant property: Since , we get: Substituting , we solve for :
Step 2: Using the Determinant Property for Since , we simplify: Substituting :
Step 3: Matching the Answer Options
The correct answer from the given choices is . Final Answer: (a) 3.