Answer:
A) Mean Slope = 8
B) Values of C; (-2.215, 4.215)
Step-by-step explanation:
A) Formula for the mean slope over an interval [a, b] is given as;
Mean slope = [f(b) - f(a)]/(b - a)
Now, for this question, we have;
f(x) = 2x³ − 6x² − 48x + 4
Thus, the mean slope of this function on interval [−4,7] will be;
[f(7) - f(-4)]/(7 - (-4))
f(7) = 2(7³) − 6(7)² − 48(7) + 4
f(7) = 60
f(-4) = 2(-4³) − 6(-4)² − 48(-4) + 4
f(-4) = -28
Thus, the mean value is now ;
[f(7) - f(-4)]/(7 - (-4)) = [60 - (-28)]/11
= 88/11 = 8
B) We seek to verify the Mean Value Theorem for the function
f(x) = 2x³ − 6x² − 48x + 4
on the interval [−4,7]
The Mean Value Theorem, tells us that if f(x) is differentiable on a interval [a,b], then ∃ c ∈ [a,b] st:
f'(c) = (f(b) − f(a)) /b − a
So, Differentiating with respect to x, we have;
f'(x) = 6x² - 12x - 48
And we seek a value c ∈[−4,7] st:
f'(c) = [f(7) − f(−4)]/(7 −(−4))
f'(c) = 6c² - 12c - 48
from earlier, f(7) = 60 and f(-4) = -28
Thus;
6c² - 12c - 48 = (60 - (-28))/11
6c² - 12c - 48 = 88/11
6c² - 12c - 48 = 8
6c² - 12c - 48 - 8 = 0
6c² - 12c - 56 = 0
Using quadratic formula to solve this, we have the roots as;
c = -2.215 or 4.215
