Answer:
a) Mean or average slope over this interval
= 8
b) f'(c) = 8
c = 4.21 or -2.21 (both in the interval given)
The values of c in the interval that whose f'(c) is equal to the average slope, 8, are 4.21 or -2.21.
Step-by-step explanation:
f(x) = 2x³ − 6x² − 48x + 9
on the interval [−4,7].
The average or mean slope over an interval [a, b] is given as
Mean slope = [f(b) - f(a)]/(b - a)
a = -4
b = 7
f(-4) = 2(-4)³ − 6(-4)² − 48(-4) + 9
= -128 - 96 + 192 + 9 = -23
f(7) = 2(7³) − 6(7²) − 48(7) + 9
= 686 - 294 - 336 + 9 = 65
Mean slope = [f(7) - f(-4)]/[7 - (-4)]
Mean slope = [65 - (-23)]/11 = (88/11) = 8
b) f'(c) = 8
f(x) = 2x³ − 6x² − 48x + 9
f'(x) = 6x² - 12x - 48
f'(c) = 6c² - 12c - 48
f'(c) = average slope = 8
6c² - 12c - 48 = 8
6c² - 12c - 56 = 0
Solving the quadratic equation
c = 4.21 or -2.21
Hence, the values of c in the interval that whose f'(c) is equal to the average slope, 8, are 4.21 or -2.21.
Hope this Helps!!!