Application Of Derivatives

Take log: Differentiate w.r.t. : Collect: So,

The slope of the curve at any point is given by the first derivative , which represents the rate of change of with respect to . To find the maximum slope, we need to maximize . Step 1: Compute the First Derivative
Differentiate with respect to : Step 2: Find the Maximum Value of the Slope
To maximize , we take the second derivative and find the critical points where the slope of is zero (i.e., where has a maximum or minimum). Differentiate with respect to : Set the second derivative equal to zero to find critical points: Step 3: Determine the Nature of the Critical Point
Use the second derivative test to check if is a maximum: - If , the function has a local maximum. - If , the function has a local minimum. Substitute into : Since at , the second derivative test is inconclusive. We need to check the first derivative around or use the first derivative test. Step 4: First Derivative Test
Evaluate around : - For : - For : - For : The calculation seems inconsistent. Let's compute correctly at : Let's try points closer to check the behavior: - For : - For : This indicates a minimum, not a maximum. Let's find the critical point by setting the first derivative to zero:
Divide by 3: Discriminant: Since the discriminant is negative, has no real roots, meaning for all (as the quadratic opens upwards and the minimum value is positive). Step 5: Re-evaluate the Maximum Slope
Since is a quadratic in with a positive leading coefficient, it has a minimum value (not a maximum) at its vertex. The vertex of occurs at : At : As , .
Let's check the behavior:
- For : - For : The slope increases without bound as increases. However, the problem asks for the maximum slope, which suggests a possible misinterpretation. Let's assume the intent is to find the maximum value of the slope function within a practical context or correct the function. The slope has no upper bound, but let's derive correctly.