Question

Simplify

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Answer 1

Answer:

f′(x)f′(x) gives you the slope of ff in x

Quite easily, if f′(x)f′(x) is positive, f(x)f(x) increases. If f′(x)f′

(x) is negative, f(x)f(x) decreases.

We know that, for y∈R∗+y∈R∗+

0<y<1⇔ln(y)<00<y<1⇔ln(y)<0

ln(1)=0ln(1)=0

1<y⇔ln(y)>01<y⇔ln(y)>0

So we can write that

f′(x)>0⇔ln(x2x+1)>0⇔x2x+1>1f′(x)>0⇔ln(x2x+1)>0⇔x2x+1>1

f′(x)<0⇔ln(x2x+1)<0⇔x2x+1<1f′(x)<0⇔ln(x2x+1)<0⇔x2x+1<1

f′(x)<0⇔ln(x2x+1)<0⇔x2x+1<1f′(x)<0⇔ln(x2x+1)<0⇔x2x+1<1If x<−1

Answer 2

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