If f(x) = |(x^2 − 4)(x^2 + 2)|, how many numbers in the interval [0, 1] satisfy the conclusion of the Mean Value Theorem?
since f(x) < 0 in the interval, we can use
f(x) = -(x^2-4)(x^2+2)
f(1) = 9
f(0) = 8
the slope of the secant is this (9-8)/(1-0) = 1
By inspection of the graph, there appear to be two values that work. But to be sure,
f'(x) = -4x(x^2-1)
So, we want to find c such that
-4c(c^2-1) = 1
solve that using your favorite tools and there are indeed two values for c.
Thank you! However, wouldn't there be three values for c?
yes, but one is not included in the interval [0,1]
Take a look at the graph here:
https://www.wolframalpha.com/input/?i=-(x%5E2+%E2%88%92+4)(x%5E2+%2B+2)
To determine how many numbers in the interval [0, 1] satisfy the conclusion of the Mean Value Theorem, we need to find if there is an instance where the instantaneous rate of change of the function is equal to its average rate of change over that interval.
The Mean Value Theorem states that if a function f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one number c in (a, b) such that the instantaneous rate of change of f(x) at c (f'(c)) is equal to the average rate of change of f(x) over the interval [a, b] (Δy/Δx).
In this case, we have function f(x) = |(x^2 − 4)(x^2 + 2)|.
First, let's find the average rate of change of f(x) over the interval [0, 1]. The average rate of change is given by (f(1) - f(0)) / (1 - 0).
Evaluate f(1):
f(1) = |(1^2 − 4)(1^2 + 2)|
= |(-3)(3)|
= 9
Evaluate f(0):
f(0) = |(0^2 − 4)(0^2 + 2)|
= |(-4)(2)|
= 8
Average rate of change = (f(1) - f(0)) / (1 - 0)
= (9 - 8) / (1 - 0)
= 1
So, the average rate of change of f(x) over the interval [0, 1] is 1.
Now, let's find the instantaneous rate of change of f(x) at any point c in the interval (0, 1).
To do this, we need to find the derivative of f(x) with respect to x (f'(x)), and then evaluate it at c.
Taking the derivative of f(x):
f(x) = |(x^2 − 4)(x^2 + 2)|
Using the product rule of differentiation:
f'(x) = (x^2 + 2) * (2x) + (x^2 - 4) * (2x)
= 2x(x^2 + 2) + 2x(x^2 - 4)
= 2x^3 + 4x + 2x^3 - 8x
= 4x^3 - 4x
Now, we need to evaluate f'(c) where c is an unknown number in the interval (0, 1).
So, f'(c) = 4c^3 - 4c
To satisfy the conclusion of the Mean Value Theorem, we need to find if there exists a value of c in (0, 1) such that f'(c) is equal to the average rate of change we found earlier, which is 1.
So, we need to solve the equation 4c^3 - 4c = 1 for c.
Simplifying the equation:
4c^3 - 4c - 1 = 0
Unfortunately, there is no algebraic way to solve this equation for c. We would need to use numerical approximation techniques (such as Newton's method or graphing) to find the value of c that satisfies the equation.
Therefore, we cannot determine how many numbers in the interval [0, 1] satisfy the conclusion of the Mean Value Theorem without further analysis or numerical approximation methods.