Answer:
To find the C-intercept of the function C(t) = 2t^4 - 14t^3 + 12t^2, you need to set C(t) equal to zero and solve for t. The C-intercept occurs when the function's value (C(t)) is equal to zero.
So, you have:
2t^4 - 14t^3 + 12t^2 = 0
Now, we can factor out the common terms:
2t^2(t^2 - 7t + 6) = 0
Now, we have a quadratic equation within the parentheses:
t^2 - 7t + 6 = 0
To solve this quadratic equation, you can use the quadratic formula:
t = (-b ± √(b² - 4ac)) / (2a)
In this case, a = 1, b = -7, and c = 6. Plug these values into the formula:
t = (-(-7) ± √((-7)² - 4(1)(6))) / (2(1))
Now, calculate the discriminant (the value inside the square root):
Discriminant = (-7)² - 4(1)(6) = 49 - 24 = 25
So, the discriminant is positive. This means there are two real roots for this quadratic equation.
Now, use the quadratic formula to find the roots:
t1 = [7 + √25] / 2 = (7 + 5) / 2 = 12 / 2 = 6
t2 = [7 - √25] / 2 = (7 - 5) / 2 = 2 / 2 = 1
Therefore, the t-intercepts of the function C(t) are t = 6 and t = 1, and the C-intercept is when C(t) = 0, which occurs at these t-values.
Explanation: