To apply the Intermediate Value Theorem, we first need to define the function based on the given equation:
f(x) = x5 – 2x4 – x – 3
Now, let’s evaluate the function at the endpoints of the interval [-2, 3].
First, we calculate f(-2):
f(-2) = (-2)5 – 2(-2)4 – (-2) – 3
f(-2) = -32 – 2(16) + 2 – 3 = -32 – 32 + 2 – 3 = -65
Next, we calculate f(3):
f(3) = (3)5 – 2(3)4 – (3) – 3
f(3) = 243 – 2(81) – 3 – 3 = 243 – 162 – 3 – 3 = 75
We have:
f(-2) = -65 and f(3) = 75
Since f(-2) is negative and f(3) is positive, by the Intermediate Value Theorem, there must be at least one root of the equation f(x) = 0 in the interval (-2, 3). This theorem states that if a continuous function takes on values of opposite signs at two points, then there exists at least one point between them where the function is zero.
Thus, we can conclude that there is a root of the equation x5 – 2x4 – x – 3 = 0 in the interval [-2, 3].