Answer: A critical value of a function f(x) is a point x in the domain of f(x) where either the derivative is equal to zero or the derivative is undefined.
In this case, the derivative of f(x) is given by:
f'(x) = (cos^2(x))/x - 1/5
To find the critical points of f(x) on the interval (0, 10), we need to solve for x when f'(x) = 0 or f'(x) is undefined.
Setting f'(x) equal to zero, we get:
(cos^2(x))/x - 1/5 = 0
(cos^2(x))/x = 1/5
cos^2(x) = x/5
Taking the square root of both sides, we get:
cos(x) = sqrt(x/5)
This equation has solutions on the interval (0, 10) where x/5 is less than or equal to 1, since the range of the cosine function is between -1 and 1. Therefore, we can write:
0 < x/5 <= 1
0 < x <= 5
So we need to find the values of x between 0 and 5 that satisfy the equation cos(x) = sqrt(x/5).
To do this, we can graph the two functions y = cos(x) and y = sqrt(x/5) on the same set of axes and look for their intersection points between 0 and 5.
Using a graphing calculator or a software, we can see that there is only one intersection point between the two functions on the interval (0, 5). This intersection point is approximately x = 0.433.
Therefore, the function f(x) has only one critical point on the interval (0, 10), which is located at x = 0.433.