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
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
To show that the equation x + ln(x + 1) = 0 has exactly one solution in the interval 0 ≤ x ≤ 3, we can follow these steps:
Step 1: Define the function f(x) = x + ln(x + 1).
Step 2: Determine the domain of the function. In this case, since x + 1 cannot be less than or equal to zero (due to the natural logarithm), we have x > -1. Therefore, the domain of f(x) is -1 < x ≤ 3.
Step 3: Show that f(x) is continuous on the interval [0, 3]. To do this, we need to confirm that f(x) is continuous at every point within the interval [0, 3]. Since f(x) is the sum of two continuous functions, x and ln(x + 1), it follows that f(x) is continuous on the entire interval [0, 3].
Step 4: Show that f(x) is monotonic on the interval. To prove that f(x) is monotonic on [0, 3], we can calculate the derivative of f(x).
f'(x) = 1 + 1/(x + 1)
Since the derivative is always positive on the interval [0, 3], it means that f(x) is strictly increasing. This implies that f(x) can have at most one solution within the interval [0, 3].
Step 5: Show that f(x) changes sign on the interval. To check for a sign change, we evaluate f(0) and f(3):
f(0) = 0 + ln(0 + 1) = ln(1) = 0
f(3) = 3 + ln(3 + 1) = 3 + ln(4) > 0
Since f(0) = 0 and f(3) > 0, it means that f(x) changes sign within the interval [0, 3]. Therefore, there exists at least one solution to the equation x + ln(x + 1) = 0.
Step 6: Combining the results from steps 4 and 5, we conclude that the equation x + ln(x + 1) = 0 has exactly one solution in the interval 0 ≤ x ≤ 3.
Please note that although we have shown that the equation has exactly one solution within this interval, we have not found the exact solution. To determine the solution, you might need to use numerical methods or further algebraic techniques.