The concentration of a certain element in the water supply of a town is modeled by the function f, where f(t) is measured in parts per billion and t is measured in years.

Question

The first derivative of f is given by f′(t)=1−ln t−sin t. At what times t, for 0 < t < 5, does the concentration attain a local minimum?

Answer 1

Eric

There will be a turning point (max or min) when f'(t)=0. So you need to solve this. Easiest way is to plot f'(t) on Desmos or a graphing calculator and zoom in on the zeroes. Make sure you are working in radians.

I just did that and there seem to be two zeros between t=0 and t=5.

Having found the zeroes you need to decide which are minima.

For this find f''(t) at each zero. If f''(t) < 0 (which means f'(t) is decreasing at the zero so going from positive to negative) then you have a maximum

If f''(t) > 0 you have a minimum (f'(t) is going from negative to positive through the zero).

To get f''(t) differentiate f'(t) once, giving -1/x - cos (t)

Mike