I need a help with this Single-Variable Optimization Problem. (Brief Calculus)?

Question

For the following function determine all numbers of x* in the specified interval such that f'(x*)=(f(b)-f(a))/(b-a) where f(x)=2/x in [2,6]. I don't know how to even start this problem.

Answer 1

a=2 and b=6 and f(x) = 2/x

f(a) = f(2) = 2/2 = 1
f(b) = f(6) = 2/6 = 1/3

f'(x*)=(f(b)-f(a))/(b-a) =
(1/3 - 1) / (6 - 2) =
-2/3 / 4 =
-1/6

Th

Answer 2

Well, you have an equation that you have to solve. Start by evaluating each side of it.

f(x) = 2/x, so f'(x) = -2/x^2 by the power rule. So f'(x*) = -2/(x*)^2.
Also, the interval is [2, 6], so a = 2 and b = 6. Hence
(f(b) - f(a)) / (b - a) = (f(6) - f(2)) / (6 - 2)
= (2/6 - 2/2) / 4
= -1/6.
So we need to find x* such that -2/(x*)^2 = -1/6
<=> (x*)^2 = 12
<=> x* = ±2√3
But -2√3 isn't in [2, 6], so we are left with
x* = 2√3.

Answer 3

After in simple terms glancing at this i might attempt to circulate it into some variety of hyperbolic functionality and a few logs ... --- i ultimately have been given this iea to artwork, yet now I see Danny852's way is plenty, plenty greater helpful!