Answer to Question #156218 in Calculus for Fletcher

Question

f is the real valued function defined by f(x) = 1 / (1 + e^x). We define the function g by g(x) = x - f(x).
(i) Draw tge variation table of g.
(ii) Show that the equation f(x) = x admits a unique solution a E ( 1/4 , 1/2 ) where E represents "an element of".

Answer 1

(i) Draw the variation table of g.

x g(x)

-9    -9.9999

-8    -8.9997

-7    -7.9991

-6    -6.9975

-5    -5.9933

-4    -4.982

-3    -3.9526

-2    -2.8808

-1    -1.7311

0    -0.5

1    0.7311

2    1.8808

3    2.9526

4    3.982

5    4.9933

6    5.9975

7    6.9991

8    7.9997

9    8.9999

(ii) Show that the equation f(x) = x admits a unique solution a E ( 1/4 , 1/2 ) where E represents "an element of".

it is a graph of f(x) = x, as we can see it intersects y=0 axes at 0.4

and 1/4 = 0.25 < 0.4 < 0.5 = 1/2

it means that the equation f(x) = x admits a unique solution a E ( 1/4 , 1/2 )