Question

evaluate (f(x) = 2x^4 - 3x^2 - 12x + 6) at (0) and (-2) to determine if the intermediate value theorem guarantees that a zero exists between the two values.

a) (f(0) =)

b) (f(-2) =)

c) does the intermediate value theorem guarantee that a zero exists between (0) and (-2)?

yes
no

Explanation:

Step1: Evaluate \(f(0)\)

Substitute \(x = 0\) into the function:

$$f(0) = 2(0)^4 - 3(0)^2 - 12(0) + 6 = 6$$

Step2: Evaluate \(f(-2)\)

Substitute \(x = -2\) into the function:

$$f(-2) = 2(-2)^4 - 3(-2)^2 - 12(-2) + 6 = 2(16) - 3(4) + 24 + 6 = 32 - 12 + 24 + 6 = 50$$

Step3: Apply Intermediate Value Theorem

Check if the function values have opposite signs:
Since both \(f(0) = 6 > 0\) and \(f(-2) = 50 > 0\) are positive, the Intermediate Value Theorem does not guarantee a zero.

Answer:

a) \(f(0) = 6\)
b) \(f(-2) = 50\)
c) no