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

0 votes

 and where is it concave up, concave down?

asked Nov 15, 2014 in PRECALCULUS by anonymous

1 Answer

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The function f(x) = 6ex/(6ex + 7)

Differentiate with respect to x.

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

f'(x) = [36e2x + 42ex - 36e2x]/(6ex + 7)2

f'(x) = 42ex/(6ex + 7)2

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

= (6ex + 7)(42ex)[(6ex + 7)(42ex) - 12ex]/(6ex + 7)4

 = 42ex[ 7 - 6ex]/(6ex + 7)3 

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

Equate secon derivative = 0.

- 42ex[ 6ex - 7]/(6ex + 7)3 = 0

- 42ex[ 6ex - 7] = 0

- 252(ex)2 + 294ex = 0

Solve the above equation using quadratic formula

image

 

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) = 6ex/(6ex + 7)

y = 6e0/(6e0 + 7)

y = 6/(6 + 7) = 0.461

f(x) = 6e1.166/(6e1.166 + 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).

answered Nov 17, 2014 by david Expert

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