Given that f'(x)=[(8cos(x)/(x2)-(⅛)], find the number of relative maxima and minima on the interval (1,10). Finding maxima and minima analytically wasn't fruitful for me, so instead I used a bit of handwaving. First I argued, using the intermediate value theorem, since f'(1)>f'(10), there exists at least one maximum or minimum. Then I said, well, the derivative is approximately equal to 8cos(x)/x2 and since cos(x) has 2 minima and one maximum on the interval (1,10), then f(x) must have 2 minima and one maximum on that same interval. Is there way to solve this without that handwaving?