Evaluate f(x)=(e^x)/[(e^x)-221]

Question

a. What is the domain of f(x)? Identify any intercepts of f(x). 

b. Identify any critical points of f(x). On what interval is f(x) increasing? On what interval is it decreasing? 

c. On what interval is f(x) concave up? On what interval is it concave down?

Answer 1

a) The function f(x) = (e^x)/(e^x - 221)

y = (e^x)/(e^x - 221)

To find which number make the fraction undefined create an equation where the denominator is not equal to zero.

e^x - 221 ≠ 0

e^x ≠ 221

log(e^x) ≠ log(221)

x log(e) ≠ log(13*17)

x ≠ log(13) + log(17)

The domain of function is real numbers except [log(3) + log(17)].

Domain set is { x ∈ R : x ≠ log(13) + log(17)}

 

To find x intercepts substitute y = 0 in y = (e^x)/(e^x - 221).

0 = (e^x)/(e^x - 221)

e^x = 0

log(e^x) = log(0)

log(0) not defined.

No solutions exist.

In this case there is no x intercept.

To find y intercepts substitute x = 0 in y = (e^x)/(e^x - 221).

y = (e⁰)/(e⁰ - 221)

y = 1/(1 - 221)

y = 1/(- 220)

y = - 0.0045

y intercept is - 0.0045.

Answer 2

c) The first derivative of f is f'(x) =  (- 221e^x )/(e^x - 221)²

Differentiating with respect to x.

f''(x) = [(e^x - 221)²(- 221e^x ) - (- 221e^x )2(e^x - 221 )(e^x)]/(e^x - 221)⁴

f''(x) = [(e^x - 221)²(- 221e^x ) + 442 (e^2x)(e^x - 221 )]/(e^x - 221)⁴

f''(x) = (e^x - 221)[(e^x - 221)(- 221e^x) + 442e^2x]/(e^x - 221)⁴

f''(x) = [- 221e^2x + 48841e^x + 442e^2x]/(e^x - 221)³

f''(x) = [ 48841e^x + 221e^2x]/(e^x - 221)³

Equate the second derivative = 0

[ 48841e^x + 221e^2x]/(e^x - 221)³ = 0

f''(x) is undefined when x = log(221).

log(221) is not in the domain of original function f(x) = (e^x)/(e^x - 221).

In this case there are no inflection points.

 

Interval    Test Value                      Sign of f''(x)                                                                

(-∞, log(221))  x = 1      f''(1) = [ 48841e¹ + 221e²⁽¹⁾]/(e¹- 221)³] = - 233406314.813 < 0      Conclusion : Concave downward.

(log(221), ∞)     x = -1     f''(10) = [ 48841e¹⁰ + 221e²⁽¹⁰⁾]/(e¹⁰- 221)³] = 1.2250585 > 0       Conclusion : Concave upward.

Answer 3

b) The function f(x) = (e^x)/(e^x - 221)

Differentiating with respect to x.

f'(x) = [(e^x - 221)((e^x) - (e^x)(e^x)]/(e^x - 221)²

f'(x) = [(e^2x - 221e^x) - (e^2x)]/(e^x - 221)²

f'(x) = (e^2x - 221e^x - e^2x)/(e^x - 221)²

f'(x) =  (- 221e^x )/(e^x - 221)²

To find critical points equate the first derivative equals to zero.

(- 221e^x )/(e^x - 221)² = 0

f'(x) is undefined when x = log(221).

log(221) is not in the domain of original function f(x) = (e^x)/(e^x - 221).

In this case there are no stationary points.

In this case the critical point is undefined value x = log(221).

The function is a monotonic function.

A monotonic function is a function which is either entirely non increasing or non decreasing.