Use the intermediate value theorem to verity that x^4+X-3=0 has a solution in the interval(1,2)
To use the intermediate value theorem to verify that the equation x^4 + x - 3 = 0 has a solution in the interval (1, 2), we need to show that the function defined by f(x) = x^4 + x - 3 takes on both positive and negative values in that interval.
To begin, substitute the lower and upper bounds of the interval into the equation:
For x = 1: f(1) = 1^4 + 1 - 3 = -1
For x = 2: f(2) = 2^4 + 2 - 3 = 15
Since f(1) = -1 and f(2) = 15, we can see that f(x) changes signs between these two endpoints of the interval. That means that at least one point between x = 1 and x = 2 must exist where f(x) = 0, satisfying the equation x^4 + x - 3 = 0.
Therefore, by the intermediate value theorem, we can conclude that the equation x^4 + x - 3 = 0 has a solution in the interval (1, 2).