Question

The geometric mean of two postitive numbers a and b is sqrt(ab).
Show that for f(x) = 1/x on any interval [a,b] of positive numbers, the value of c in the conclusion of the mean value theorem is c = sqrt(ab)

I have no idea how to do this!

If the mean of a and b is sqrt(ab), then (a + b) / 2 = sqrt(ab).

Is c supposed to be the area under the graph in interval [a,b] ? If so, calculate the antiderivative of f(x) and you'll have the formula for c.

It is the geometric mean. Isn't (a+b)/2 the arithmetic mean?

Answer 1

etyhw

Answer 2

need the geometric mean for 4 and 6

Answer 3

f(b)-f(a)
________ = f'(c)
b-a

1/b - 1/a
__________ = -1/(c)^2
b-a