We start by differentiating, using the product rule and the chain rule.
Let y = u^(1/2) and u = x.
y' = 1/(2u^(1/2)) and u' =1
y' = 1/(2(x)^(1/2))
Now, by the product rule;
f'(x) = 0 xx sqrt(x) + 1/(2(x)^(1/2)) xx 5/2
f'(x) = 5/(4sqrt(x))
The rate of change at any given point on the function is given by evaluating x = a into the derivative. The question says that the rate of change at x = 3 is twice the rate of change at x = c. Our first order of business is to find the rate of change at x = 3.
r.c = 5/(4sqrt(3))
The rate of change at x = c is then 10/(4sqrt(3)) = 5/(2sqrt(3)).
5/(2sqrt(3)) = 5/(4sqrt(x))
20sqrt(x) = 10sqrt(3)
20sqrt(x) - 10sqrt(3) = 0
10(2sqrt(x) - sqrt(3)) = 0
2sqrt(x) - sqrt(3) = 0
2sqrt(x)= sqrt(3)
4x = 3
x = 3/4
So, the value of c is 3/4.
Hopefully this helps!
