The graph of the marginal revenue function from the sale of...

The graph of the marginal revenue function from the sale of $x$ sports watches is given in the figure. (A) Using the graph shown, describe the shape of the graph of the revenue function $R(x)$ as $x$ increases from 0 to 1,000 (B) Find the equation of the marginal revenue function (the linear function shown in the figure). (C) Find the equation of the revenue function that satisfies $R(0)=0$. Graph the revenue function over the interva $[0,1,000] .[$ Check the shape of the graph relative to the analysis in part (A). (D) Find the price-demand equation and determine the price when the demand is 700 units.

Transcript

00:01 Hello everyone, we are going to solve a question.

00:03 In a part of this question, we have to find shape of revenue function.

00:18 So from here we have if r -dash -x is positive, if this is positive, then implies r -x is increasing.

00:35 If r -dash -x is negative, so that means this.

00:43 Implies r of x is decreasing so since r x is sorry r d 'ex is is linear function so from here we have that r of x is quadratic if this is quadratic that means which is the shape is parabolic so from here we have that since from here we have therefore r of x must be an opening downward downwards parabola increasing up to increasing up to x equals to 500 and decreasing in x from 500 to thousand so that means this is the final answer of a part of the question moving towards the b part of the question in b part we have to find the equation of equation of marginal revenue equation or we can say equation of marginal revenue only.

02:36 So from here the solution is as we know, y is equals to mx plus c.

02:44 So from here, m is y2 minus y1 divided by x to minus x1.

02:51 So for this equation, we can find m's law pass.

02:55 0 minus 100 divided by 500 minus 0 that is equals to minus 1 divided by 5.

03:03 So now c is intercept.

03:05 So from here c is equals to 100.

03:08 So that means equation of, equation of marginal revenue equation of marginal revenue function is given by r -dash -x equals to minus 1 divided by 5x plus 100.

03:35 This is the answer for d part of the question.

03:39 Moving towards the c part, in c -part, we have to find the equation that satisfies the r0 equals to 0 and we have to make a graph for this also.

03:59 So from here we have r dash x is equals to minus x divided by 5 plus 100.

04:08 On integrating we have integration of r dash x is equals to integration of minus x divided by 5 plus 100 with respect to dx.

04:19 On solving this we have r of x is equals to minus x squared divided by 10 plus 100 x plus c.

04:29 So from here we are given that r of zero equals to zero.

04:35 So this implies c is equals to zero.

04:38 So that means r of x is equals to minus x squared divided by 10 plus 100 x...

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