To satisfy the mean value theorem, we first need to check if the function f(x) is continuous on the interval [-2, 1] and differentiable on the open interval (-2, 1).
The function f(x) = x^3 + x + 1 is a polynomial function. Polynomials are continuous and differentiable for all real numbers. Therefore, f(x) is continuous on the interval [-2, 1] and differentiable on the open interval (-2, 1).
Now, let's find the average rate of change of f(x) over the interval [-2, 1] using the mean value theorem.
The average rate of change of f(x) over the interval [-2, 1] is given by:
f'(c) = (f(1) - f(-2))/(1 - (-2))
To find c, we need to find the derivative of f(x) and evaluate it at some point in the open interval (-2, 1).
Taking the derivative of f(x), we get:
f'(x) = 3x^2 + 1
Now, we need to evaluate f'(x) at c:
3c^2 + 1 = (f(1) - f(-2))/3
We can substitute the x-values of 1 and -2 into f(x):
3c^2 + 1 = ((1)^3 + (1) + 1 - ((-2)^3 + (-2) + 1))/3
3c^2 + 1 = (1 + 1 + 1 - (-8 - 2 + 1))/3
3c^2 + 1 = (3 - 5)/3
3c^2 + 1 = -2/3
Subtracting 1 from both sides:
3c^2 = -2/3 - 1
3c^2 = -2/3 - 3/3
3c^2 = -5/3
c^2 = -5/9
Since c^2 cannot be negative for real numbers, there is no value of c in the interval [-2, 1] that satisfies the mean value theorem for the function f(x) = x^3 + x + 1.
Find c in the interval [-2,1] which satisfies the mean value theorem for f(x)=x^3+x+1
1 answer