Application Of Derivatives

The volume of a sphere is given by the formula: where is the radius of the sphere. We are given that the diameter is increasing at a rate of 3 cm/min. Since the radius is half of the diameter, the rate of change of the radius is: Now, differentiate the volume equation with respect to time : Substitute the given values: when cm, the radius cm. We already know that cm/min. Thus, the rate of change of the volume is:

So, the correct option is (B) :

Step 1: Identify the function and interval: We need to find its minimum on .

Step 2: Analyze the function behavior: The expression under the square root must be non-negative: This holds for all .

Step 3: Find critical points by taking derivative: Set :

Step 4: Evaluate function at critical point and endpoints:

Step 5: Determine the minimum:

The smallest value among these is .

Given:

Then ⇒ at

means the slope

Therefore, the equation of the line passing through origin which is parallel to the tangent of the curve at is