How to use the intermediate value theorem?

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E^(5*-2) - 2 0 Since f(x) is continuous in the interval -2,2 and f(-2) 0, then there must be some point -2.

Set f(x) = e^(5x) + x. First, we see that f(x) is continuous. We need to find a value x1 so that f(x1) 0.

I'll leave that to you. Thus, by IVT, because f(x1).

When x is a large negative number the exponential is small and the expression on the left is dominated by the x term, hence the expression will also be a large negative number. Likewise, when x is a large positive number both the exponential and the x term are large positive, so the expression takes on a large positive value. Lastly, the expression is the sum of two continuous functions and thus itself is continuous.So what we have is a continuous function that is negative somewhere and positive somewhere else, hence it must pass through zero so there must be a solution to the original equation.

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