F(x)=1/3 t^3-5/2 t^2+80;(0=t =10) Find the intervals where the function F(t) is increasing and the intervals where it is decreasing.

Question

The Department of Environment and Natural Resources began to record an index of environment quality that measured progress and decline in the environmental quality of its forests. The index for the years 1998 through 2008 is approximately by the function:
F(x)=1/3 t^3-5/2 t^2+80;(0≤t ≤10)
Where t = 0 corresponds to 1998. Find the intervals where the function F(t) is increasing and the intervals where it is decreasing. Interpret your results (in no more than 2 sentences).

Answer 1

Find the derivative of F 

 

F'(t) = t^2 - 5t = t(t-5) 

 

F(t) is increasing where the derivative is positive and is decreasing where the derivative is negative.  

 

So F(t) is decreasing when 0<t<5 

and F(t) is increasing when 5<t<10