Find the minimum and maximum of the function z = 4x

Find the minimum and maximum of the function $z = 4x - 4y$ subject to \begin{cases} y \ge 3\\y \le 7\\4x - 3y \ge -9\\4x + y \le 39 \end{cases} What are the corner points of the feasible set? The minimum is and maximum is 24. Type \text{``None''} in the blank provided if the quantity does not exist.

Transcript

00:01 Hello, as in discussion we have to maximize, maximize p is equals to 7x minus of 5y subject to constraint are given as here x is greater than equals to 0, y is greater than equals to 0, 3x plus 6y less than equals to 18 and this 4x plus 4y greater than equals to 16.

00:22 So firstly we have to find the corner points of the feasible set then we have to find maximum value and point at which maximum value occurs.

00:31 So firstly if we write here these are non -negative quantities here and if we solve these inequalities, firstly we solve these inequalities in equation so we can write it as here 3x plus 6y equals to 18.

00:44 By simplifying this equation we can write it as here x plus 2y is equals to here 6.

00:51 So if we put here x as 0, if x as 0 then why we get 2y equals to 6 which is equals to 3 and whenever y equals to 0, x equals to 6.

01:01 Similarly if we solve another inequality which is 4x plus 4y, this one is 4y here, 4y is equals to 16 here so by simplifying this inequality we can write it as here x plus y is equals to 4 here so our points are when x equals to 0, y equals to 4, when y equals to 0, x equals to 4 here.

01:25 So if we plot here this one is let us take our x axis respectively in the perpendicular sense this one is our y axis.

01:32 If we plot our this line 0, 4 and 4, 0 this one will be our line like this here and for the another one as we can see here our another line for 0, 3 and minus 6 as this one here, this one is our another line.

01:49 As we can see in the first term, in the first inequality we plot 0, 0 point as 0 is less than 18 so inequality satisfy so this one is implying origin is in the reason here.

02:01 So for this line here, this one is our point 3 comma as this one is our point is here.

02:11 Let us take this one is x axis, this one is y axis, this point is 4 here, this point is 6 here respectively this point is 3 here and this point is 4 here, this point is origin here.

02:21 In the first case origin is in the reason so our reason will be in this sense here.

02:27 This one will be our reason as x is greater than 0 and y is greater than 0.

02:32 This one is implying here our reason in first quadrant here from the these inequalities and this one in this way is our reason.

02:41 If we take here in the second case 0 greater than 16 this condition is not possible.

02:46 If this one is not possible then reason will be to upward from origin so this one will be reason.

02:53 If we take here our intersecting reason will be this one as this one is our intersecting reason so these are our corner points.

03:01 Let us take this point is a here, this point is b here and this point is c here...

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