Answer to Question #156221 in Calculus for Rashinda

Question #156221
Given that f is the real valued function defined by f(x) = 1 / (1 + e^x). And I = (1/4, 1/2).
(i)Show that if x E I, then f(x) E I. Where E represents "an element of".
(ii) Calculate f''(x), hence show that if x E I, thenㅣ f'(x)ㅣ<= 1/4.
(iii) Deduce that if x E I, then ㅣf(x) - sㅣ<= 1/4 ㅣx - sㅣ
s E (1/4, 1/2).
1
Expert's answer
2021-01-26T04:33:46-0500

(i) xI,I=(14, 12)x\in I, I=(\dfrac{1}{4}, \ \dfrac{1}{2})


f(x)=11+exf(x)=\dfrac{1}{1+e^x}

f(x)=(11+ex)=ex(1+ex)2<0,xRf'(x)=(\dfrac{1}{1+e^x})'=-\dfrac{e^x}{(1+e^x)^2}<0, x\in \R

Therefore the function f(x)f(x) decreases for xR.x\in \R.


f(14)=11+e1/4<11+(1)1/4=11+1=12f(\dfrac{1}{4})=\dfrac{1}{1+e^{1/4}}<\dfrac{1}{1+(1)^{1/4}}=\dfrac{1}{1+1}=\dfrac{1}{2}

f(12)=11+e1/2>11+(4)1/2=11+2=13>14f(\dfrac{1}{2})=\dfrac{1}{1+e^{1/2}}>\dfrac{1}{1+(4)^{1/2}}=\dfrac{1}{1+2}=\dfrac{1}{3}>\dfrac{1}{4}

Therefore


14<f(x)<12,x(14, 12)\dfrac{1}{4}<f(x)<\dfrac{1}{2}, x\in(\dfrac{1}{4},\ \dfrac{1}{2})

Therefore if xI,x\in I, then f(x)I.f(x)\in I.


(ii)


f(x)=(11+ex)=ex(1+ex)2<0,xRf'(x)=(\dfrac{1}{1+e^x})'=-\dfrac{e^x}{(1+e^x)^2}<0, x\in \R

f(x)=(ex(1+ex)2)f''(x)=(-\dfrac{e^x}{(1+e^x)^2})'

=ex(1+ex)22e2x(1+ex)(1+ex)4=-\dfrac{e^x(1+e^x)^2-2e^{2x}(1+e^x)}{(1+e^x)^4}

=ex(ex1)(1+ex)3>0,x>0=\dfrac{e^x(e^x-1)}{(1+e^x)^3}>0, x>0

Therefore for x>0x>0 the function f(x)f'(x) increases and f(x)<0.f(x)<0.


Therefore for x>0x>0 the function f(x)|f'(x) | decreases and f(x)>0.|f(x)|>0.


f(x)<f(0)=e0(1+e0)2=14,xI|f'(x)|<|f'(0)|=\dfrac{e^{0}}{(1+e^{0})^2}=\dfrac{1}{4}, x\in I

f(x)14,xI|f'(x)|\leq\dfrac{1}{4}, x\in I

(iii)

Let x=0.3,s=0.3x=0.3, s=0.3


f(0.3)=11+e0.3>11+1.5=0.4f(0.3)=\dfrac{1}{1+e^{0.3}}>\dfrac{1}{1+1.5}=0.4


f(x)s>0.40.3=0.1|f(x)-s|>|0.4-0.3|=0.1

14xs=140.30.3=0\dfrac{1}{4}|x-s|=\dfrac{1}{4}|0.3-0.3|=0

Hence the statement


f(x)s14xs,s(14, 12)|f(x)-s|\leq\dfrac{1}{4}|x-s|, s\in(\dfrac{1}{4}, \ \dfrac{1}{2})

is False.



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