Using the mean value theorem;
F'(x) = f(b)-f(a) / b-a
f(x)=x^2-8x+3; interval [-1,6]
I assume you want to find c such that f'(c) = (f(6)-f(-1))/7
nothing simpler:
f'(x) = 2x-8
f(6) = -9
f(-1) = 12
so, we want f'(c) = -21/7 = -3
2x-8 = -3
x = 5/2
To find the average rate of change of a function over an interval using the mean value theorem, we need to find the derivative of the function and evaluate it at a specific point within the interval.
First, let's find the derivative of the function f(x) = x^2 - 8x + 3.
f'(x) = 2x - 8
Now, let's determine the values for a and b from the given interval [-1, 6].
a = -1 and b = 6
Now, we can use the mean value theorem formula to find the average rate of change:
F'(x) = f(b) - f(a) / b - a
F'(x) = f(6) - f(-1) / 6 - (-1)
To compute f(6), substitute x = 6 into the original function:
f(6) = (6)^2 - 8(6) + 3
= 36 - 48 + 3
= -9
Similarly, to compute f(-1), substitute x = -1 into the original function:
f(-1) = (-1)^2 - 8(-1) + 3
= 1 + 8 + 3
= 12
Substituting these values into the formula:
F'(x) = (-9) - 12 / 6 - (-1)
= -21 / 7
= -3
Therefore, using the mean value theorem, the average rate of change of the function f(x) = x^2 - 8x + 3 over the interval [-1, 6] is -3.