 Hi and welcome to the session. I am Shashi and I am going to help you with the following question. The question is, find the maximum profit that a company can make if the profit function is given by px equal to 41 minus 24x minus 18x square. First of all, let us understand that if we are given function f, define our interval i, c is any point belonging to interval i such that f double dash c exists, then c is a point of local maxima if f dash c is equal to 0 and f double dash c is less than 0 and c is a point of local minima if f dash c is equal to 0 and f double dash c is greater than 0. This is the key idea to solve the given question. Now, let us start the solution. We have given profit function p is given by px equal to 41 minus 24x minus 18x square. Differentiating both sides with respect to x, we get p dash x equal to 0 minus 24 minus 36x. We know derivative of 41 is 0, derivative of 24x is 24 and derivative of 18x square is 36x. So, we get p dash x equal to minus 24 minus 36x. Now, let us find the points at which p dash x is equal to 0, p dash x is equal to 0 implies minus 34 minus 36x is equal to 0. Now, adding 24 on both sides, we get minus 36x is equal to 24. Now, dividing both sides by minus 36, we get x equal to 24 upon minus 36. Now, this implies x is equal to minus 2 upon 3. Now, we know p dash x is equal to minus 34 minus 36x. Now, again differentiating both sides with respect to x, we get p double dash x equal to minus 36. Now, we know at x equal to minus 2 upon 3, p dash x is equal to 0 and p double dash x is equal to minus 36 which is less than 0. So, this implies x equal to minus 2 upon 3 is the point of local maximum of p. Now, we get maximum profit occurs at x equal to minus 2 upon 3. Value of maximum profit is given by p minus 2 upon 3 which is equal to 41 minus 24 multiplied by minus 2 upon 3 minus 18 multiplied by minus 2 upon 3 square. Now, this is equal to 41 plus 48 upon 3. We know minus 24 multiplied by minus 2 upon 3 is equal to 48 upon 3 minus 18 multiplied by 4 upon 9 square of minus 2 upon 3 is 4 upon 9. Now, we get 41 plus 16. We know 48 divided by 3 is equal to 16 minus 8. 18 multiplied by 4 upon 9 is equal to 8. So, we can write 8 here. Now, simplifying, we get p minus 2 upon 3 equal to 49. So, maximum profit that a company can make is equal to 49 unit. So, the required answer is maximum profit is equal to 49 unit. This completes the session. Hope you understood the session. Take care and keep learning.