Question

Select all intervals in which a real zero is located for the function f(x)= x⁴ – 2x³+ 3x²-⁵

A) x = –2 and x = -1
B) x = -1 and x = 0
C) x = 0 and x = 1
(D) x = 1 and x = 2
E) X = 2 and x = 3
F) x = 3 and x = 4

Answer 1

To find the intervals in which a real zero is located for the function f(x)= x⁴ – 2x³+ 3x²-⁵, we need to analyze the signs of the function in different intervals. The intervals in which a real zero is located are -1 < x < 0 and 0 < x < 1.

Step-by-step explanation:

The function f(x)= x⁴ – 2x³+ 3x²-⁵ is a polynomial function. To find the intervals in which a real zero is located, we need to analyze the signs of the function in different intervals.

1. Let's consider the interval x < -1. Plugging in a value less than -1, such as -2, we get f(-2) = (-2)⁴ – 2(-2)³+ 3(-2)²-⁵ = 16 + 16 + 12 - 5 = 39. Since the result is positive, there is no real zero in this interval.
2. Now let's consider the interval -1 < x < 0. Plugging in a value in this range, such as -0.5, we get f(-0.5) = (-0.5)⁴ – 2(-0.5)³+ 3(-0.5)²-⁵ = 0.0625 - 0.5 + 0.375 - 5 = -5.0625. Since the result is negative, there is a real zero in this interval.
3. Next, let's consider the interval 0 < x < 1. Plugging in a value in this range, such as 0.5, we get f(0.5) = (0.5)⁴ – 2(0.5)³+ 3(0.5)²-⁵ = 0.0625 - 0.5 + 0.375 - 5 = -5.0625. Since the result is negative, there is a real zero in this interval.
4. Finally, let's consider the interval x > 1. Plugging in a value greater than 1, such as 2, we get f(2) = 2⁴ – 2(2)³+ 3(2)²-⁵ = 16 - 16 + 12 - 5 = 7. Since the result is positive, there is no real zero in this interval.

Therefore, the intervals in which a real zero is located are -1 < x < 0 and 0 < x < 1.