Maxima And Minima

Step 1: Calculate the price at
The price is given by: Step 2: Compute the rebate
The original price is Rs. 350. The rebate is: Step 3: Final result
The rebate required to maximise the revenue is Rs. 125 per calculator.

Step 1: Express revenue as a function of
Revenue is given by: Step 2: Differentiate to find critical points
The first derivative of is: For maximum or minimum, set : Step 3: Verify using the second derivative
The second derivative of is: Since , is maximum when . Step 4: Final result
The number of units that should be sold to maximise revenue is .

To find the point where the slope is maximum, we need to take the derivative of the function with respect to and find the critical points. First, compute : Next, set the first derivative equal to zero to find the critical points: Solving this quadratic equation gives: Substituting into the original equation to find the corresponding -value: Hence, the slope is maximum at .

We are given the volume function: Differentiate with respect to : Now, set to find the value of that maximizes : Thus, the volume is maximized when: