Thanks! Consider the function f(x) = 6 - 8x2 on the interval [-4, 7]. Find the a

Question

Thanks!

Consider the function f(x) = 6 - 8x2 on the interval [-4, 7]. Find the average or mean slope of the function on this interval, i.e. f(7) - f(-4)/7 - (-4) = By the Mean Value Theorem, we know there exists a C in the open interval (-4, 7) such that f'(c) is equal to this mean slope. For this problem, there is only one c that works. Find it. Consider the function f(x) = 1/x on the interval [1, 12]. Find the average or mean slope of the function on this interval. By the Mean Value Theorem, we know there exists a c in the open interval (1, 12) such that f'(c) is equal to this mean slope. For this problem, there is only one c that works. Find it.

Explanation / Answer

f(7) - f(-4) / 7-(-4) = (6-8*7*7)- (6-8(-4)(-4) )/ 11 = -386-(-122)/11 = 122-386/11 = -264/11 = -24

f'(x) = -16x as f(x) = 6-8x^2

==> f'(c) = -16c

f'(c) =-24

-16c = -24

c = 24/16 = 6/4 = 3/2 = 1.5

c= 1.5

b)mean slope = 1/12 -1/ 12-1 = 1-12/11*12 = -11/11*12 = -1/12 = -0.08333

f'(x) = -1/x^2

f'(c) = -1/c^2 = -1/12

c^2 = 12

c= 2sqrt(3)... or -2(sqrt(3)

but our interval = [1,12] ..

sO c = 2sqrt(3) = 3.464

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