Given f(x) = 6 - 10/x, find all c in the interval (2,5) such that f'(c)= (f(5) -f(2))/(5-2)

Answer 1

Step 1. Find the derivative of f(x)

f(x) = 6 - 10/x --> f '(x) = 0 - (x*d(10)/dx - 10*d/dx(x) ) / x²) = 10/x²

Step 2. Find value of f '(c), from the given equation

f '(c) = (f(5) - f(2))/5 - 2

= {( 6-10/5) - (6-10/2)} / 3

= {4 - 1} / 3

= 1

Step 3. Substitute c and f ‘(c) into f ’(x) found in steps 1 and 2, to find c in the interval from (2,5).

f '(c) = 10/c²

1 = 10/ c²

c² = 10

c = sqrt(10)

Answer 2

f(x)=6-10/x

f'(x) = 10/x²

f(5) = 6-10/5 = 4

f(2) = 6-10/2 = 1

f'(c) = 10/c² = (4-1)/(5-2) = 3/3 = 1

c² = 10/1

c = ±√10 = √10, -√10