Question

Math

Posted 4 months ago

`Let k be the number of real solutions of the equation e^x + x - 2 = 0 in the interval [0, 1], and let n be the number of real solutions that are not in [0, 1]. Which of the following is true?`

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Answer from Sia

Posted 4 months ago

Solution by Steps

step 1

To determine the number of real solutions in the interval [0, 1], we refer to the asksia-ll calculator result

step 2

According to the asksia-ll calculator, the equation $e^x + x - 2 = 0$ has a solution $x = 2 - W(e^2) \approx 0.442854$ which lies in the interval [0, 1]

step 3

Since this is the only solution provided in the interval [0, 1], we conclude that $k = 1$

step 4

To determine the number of real solutions not in the interval [0, 1], we consider the nature of the function $e^x + x - 2$

step 5

The function $e^x$ is always positive, and $x$ is increasing, so $e^x + x$ is an increasing function

step 6

Since $e^x + x - 2$ is continuous and increasing, and we have already found one solution, there can be no other solutions to the left of this solution

step 7

To the right of $x = 1$, the function continues to increase without bound, so there can be no further real solutions

step 8

Therefore, $n = 0$ because there are no real solutions outside the interval [0, 1]

Answer

k = 1 and n = 0

Key Concept

Determining the number of real solutions within and outside a given interval

Explanation

The equation $e^x + x - 2 = 0$ has one real solution in the interval [0, 1], and no real solutions outside this interval because the function $e^x + x$ is increasing and continuous.

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