The total worldwide box-office receipts for a long-running movie are approximated by the function T(x) = 100x^2 / x^2 + 25

Question

where T (x) is measured in millions of dollars and x is the number of years since the movie’s released. Sketch the graph of the function T and interpret your results. find critical points, the intervals of increase and decrease, asymptotes, and inflection points (if any) to get the graph.

Answer 1

I'm assuming you mean:

The graph, of course, looks like this:

To find the critical points, take the derivative and set it equal to xero.

To take the derivative, use the quotient rule:

(u/v)' = (u'v - uv')/v²

Where u = 100x² meaning u' = 200x

And v = x² + 25 meaning v' = 2x and v² = (x² + 25)²

So:

T'(x) = [(200x)(x² + 25) - (100x²)(2x)]/(x² + 25)²

T'(x) = [200x³ + 5000x - 200x³]/(x² + 25)²

T'(x) = 5000x/(x² + 25)²

Setting equal to zero:

5000x/(x² + 25)² = 0

Since the denominator cannot contribute to the left side of the equation being equal to zero, we can ignore it:

5000x = 0

x = 0

The only critical point is x = 0

The derivative is positive for x > 0 therefore the function is increasing on (0, ∞)

There is no reason to consider the negative portion of the graph (x < 0) since x is years since the movie's release (but if you did, T' < 0 for x < 0 so the function decreases there).

The horizontal asymptote is y = 100 meaning the the limit as x approaches infinity is 100 meaning the film will approach $100 million in box office receipts over time.

To find points of inflection, take the second derivative (also using the quotient rule):

T''(x) = (-5000)(3x² - 25)/(x² + 25)³

Set it equal to zero to get:

3x² - 25 = 0

3x² = 25

x² = 25/3 so x = ± 5/√3 and again, we can ignore the negative result since x < 0 is meaningless in this case.

So there is a point of inflection at x = 5/√3 or (rationalizing) at x = 5√3/3 (approx x = 2.89) which is the place where the rate of ticket sales begins to slow down (between x = 0 and x = 2.89 the box office receipts are accelerating (the rate is increasing). After that, despite the fact that the box office receipts are still increasing, their rate of increase is decreasing.