Application Of Derivatives

Step 1: Find the first derivative of the function.
To find the local maxima and minima, we first find the first derivative of the function :

Step 2: Set the first derivative equal to zero.
To find the critical points, we solve : Factor out : Now, solve the quadratic equation : Thus, .

Step 3: Use the second derivative to determine the nature of the critical points.
The second derivative of is:

Step 4: Check the values of at the critical points.
- At : - At : - At :

Step 5: Conclusion.
- Local maximum occurs at with . - Local minima occur at with and at with .

Step 1: Find the first derivative of the function.
The first derivative of the function is:

Step 2: Find when the derivative is negative.
For the function to be decreasing, we need . So, solve:

Step 3: Conclusion.
The function is decreasing for .