Let F(x) = square root of X and [a,b]. Find all numbers c in (a,b) such that
f'(c) = f(b)-f(a)/b-a
To find the numbers c in the interval (a, b) that satisfy the equation f'(c) = (f(b) - f(a))/(b - a), we need to follow these steps:
1. Calculate f'(x), the derivative of the function f(x) = √x. The derivative of f(x) can be found using the power rule of differentiation, which states that if f(x) = x^n, then f'(x) = n*x^(n-1). Applying this rule to our function, we have:
f'(x) = (1/2)*x^(-1/2)
2. Now, let's compute the value of f'(c) using the derivative expression from step 1. Substitute c into the expression for f'(x):
f'(c) = (1/2)*c^(-1/2)
3. Next, calculate the value of (f(b) - f(a))/(b - a). Substitute the values of f(b) = √b and f(a) = √a, and use the given interval [a, b] to compute:
(f(b) - f(a))/(b - a) = (√b - √a)/(b - a)
4. Now, we can set up the equation f'(c) = (f(b) - f(a))/(b - a) and substitute the expressions from steps 2 and 3:
(1/2)*c^(-1/2) = (√b - √a)/(b - a)
5. Simplify the equation by multiplying both sides by 2 to get rid of the fraction:
c^(-1/2) = 2*(√b - √a)/(b - a)
6. Take the reciprocal of both sides of the equation to isolate c:
c^(1/2) = (b - a)/(2*(√b - √a))
7. Finally, square both sides of the equation to solve for c:
c = [(b - a)/(2*(√b - √a))]^2
The resulting equation will give you the values of c that satisfy f'(c) = (f(b) - f(a))/(b - a) in the interval (a, b).