Where are the inflection points of the function (6e^x)/(6e^x+7),

Question

 and where is it concave up, concave down?

Answer 1

The function f(x) = 6e^x/(6e^x + 7)

Differentiate with respect to x.

f'(x) = [(6e^x + 7)(6e^x) - (6e^x)(6e^x)]/(6e^x + 7)²

f'(x) = [36e^2x + 42e^x - 36e^2x]/(6e^x + 7)²

f'(x) = 42e^x/(6e^x + 7)²

f''(x) = [(6e^x + 7)²(42e^x) - (42e^x)12e^x(6e^x + 7) ]/(6e^x + 7)⁴

= (6e^x + 7)(42e^x)[(6e^x + 7)(42e^x) - 12e^x]/(6e^x + 7)⁴

  = 42e^x[ 7 - 6e^x]/(6e^x + 7)³  

f''(x) = - 42e^x[ 6e^x - 7]/(6e^x + 7)³

Equate secon derivative = 0.

- 42e^x[ 6e^x - 7]/(6e^x + 7)³ = 0

- 42e^x[ 6e^x - 7] = 0

- 252(e^x)² + 294e^x = 0

Solve the above equation using quadratic formula

 

The test intervals are (-∞, 0), (0, 1.166) and (1.166, ∞).

Interval    Test Value                     Sign of f''(x)                                       Conclusion

(-∞, 0)      x = -1    f''(- 1) = - 42e^(-1)[ 6e^(-1) - 7]/(6e^(-1)+ 7)³ = 0.094 > 0 Concave upward.

(0, 1.166)  x = 1   f''(1) = - 42e⁽¹⁾[ 6e⁽¹⁾ - 7]/(6e⁽¹⁾+ 7)³ = - 0.0839 < 0   Concave downward.

(1.166, ∞)  x = 2    f''(2) = - 42e⁽²⁾[ 6e⁽²⁾ - 7]/(6e⁽²⁾+ 7)³ = - 0.0856 < 0  Concave downward.

 

To find inflection points, using the x - values find the corresponding y - value with the curve.

Now Substitute the values x = 0 and 1.166 in original function.

f(x) = 6e^x/(6e^x + 7)

y = 6e⁰/(6e⁰ + 7)

y = 6/(6 + 7) = 0.461

f(x) = 6e^1.166/(6e^1.166 + 7)

y = [6(3.209)]/[6(3.209) + 7] = 0.7333

Points of inflections are (x, y) = (0, 0.461) and (1.166, 0.733).