(i) x∈I,I=(41, 21)
f(x)=1+ex1
f′(x)=(1+ex1)′=−(1+ex)2ex<0,x∈R Therefore the function f(x) decreases for x∈R.
f(41)=1+e1/41<1+(1)1/41=1+11=21
f(21)=1+e1/21>1+(4)1/21=1+21=31>41 Therefore
41<f(x)<21,x∈(41, 21)Therefore if x∈I, then f(x)∈I.
(ii)
f′(x)=(1+ex1)′=−(1+ex)2ex<0,x∈R
f′′(x)=(−(1+ex)2ex)′
=−(1+ex)4ex(1+ex)2−2e2x(1+ex)
=(1+ex)3ex(ex−1)>0,x>0Therefore for x>0 the function f′(x) increases and f(x)<0.
Therefore for x>0 the function ∣f′(x)∣ decreases and ∣f(x)∣>0.
∣f′(x)∣<∣f′(0)∣=(1+e0)2e0=41,x∈I
∣f′(x)∣≤41,x∈I(iii)
Let x=0.3,s=0.3
f(0.3)=1+e0.31>1+1.51=0.4
∣f(x)−s∣>∣0.4−0.3∣=0.1
41∣x−s∣=41∣0.3−0.3∣=0 Hence the statement
∣f(x)−s∣≤41∣x−s∣,s∈(41, 21) is False.
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