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Question
Let f be the function defined by f(x)=(x^3)+a(x^2)+bx+c and having the following properties.
(i) The graph of f has a point of inflection at (0,-2).
(ii) The average (mean) Value of f(x) on the closed interval [0,2] is -3.
(a) Determine the values of a, b, and c.
(b) Determine the value of x that satisfies the conclusion of the Mean Value Theorem for f on the closed interval [0,3].
(i) A point of inflection is where f"(x) has the value 0.
f"(x) = 6x + 2a. For the point (0,-2) to make this equation true,
just solve 6(0) + 2a = -2.
(ii) Once a has been solved for and put into the equation,
integrate the equation from 0 to 2, so find what
⌠2
⌡0 x^3 - x² + bx + c dx is equal to.
Divide this by the interval width to get the average value.
The interval width is 2 - 0 = 2, and that value is -3.
The value of the integral is (x^4/4 - x^3/3 + bx²/2 + cx)
from 0 to 2, which is (4 - 8/3 + 4b/2 + 2c) - 0 = 4/3 + 2b + 2c.
If the average is -3, we need to divide that value by 2, giving
2/3 + b + c = -3.
(a) We don't have enough data to determine a, b, and c.
To determine three variables, you need three equations, but we only have two of them here. We do have enough data to determine a, but what we can say about b and c is that b + c = -11/3.
(b) To determine x, we need first to have enough info to compute a and b.
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