use intermediate value theorem to show f(x) has a zero
f(x)= x^5 - 4x^4- 7x^2 - 6x; [-0.7, -0.6]
To use the Intermediate Value Theorem, we need to check whether the function f(x) changes sign between the interval [-0.7, -0.6]. If f(x) changes sign, this means that f(x) has a zero within the interval.
First, we need to evaluate f(-0.7) and f(-0.6) to see if the signs of these two function values are different.
Let's calculate f(-0.7):
f(-0.7) = (-0.7)^5 - 4(-0.7)^4 - 7(-0.7)^2 - 6(-0.7)
= -0.16807 - 0.41184 - 0.343 - (-4.2)
= -0.16807 + 0.41184 - 0.343 + 4.2
= 4.1 - 0.16807 - 0.343
= 3.58893
Now let's calculate f(-0.6):
f(-0.6) = (-0.6)^5 - 4(-0.6)^4 - 7(-0.6)^2 - 6(-0.6)
= -0.07776 - 0.20736 - 0.252 - (-3.6)
= -0.07776 + 0.20736 - 0.252 + 3.6
= 3.88 - 0.07776 - 0.252
= 3.55
Since f(-0.7) = 3.58893 and f(-0.6) = 3.55, we can see that the function values have the same sign. Therefore, the Intermediate Value Theorem does not apply in this case, and we cannot conclude that f(x) has a zero within the interval [-0.7, -0.6].