Calculas HS

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show that the solution has exactly one solution in the interval

x+ln(x+1)=0, 0<and equal to x <and equal to 3

• Calculas HS -

x+ln(x+1)=0 , 0 ≤ x ≤ 3

write your equation as ln(x+1) = -x
now let the left side by
y = ln(x+1) and the right side
y = -x

and make a quick sketch of each on the same x-y plane.
the first one is the log function crossing the x axis at the origin, slowing rising in the first quadrant, and dropping down into the third quadrant.
the other graph is a straigh line with slope of -1 and passing through the origin.

Clearly they can only cross at x=0, which becomes your only answer.

check : if x=0
LS = 0 + ln(0+1)
= 0 + 0 = 0 = RS

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