Question

find the intervals in which f(x)=3x^4/10-4x^3/5-3x^2+36x?5+11

Answer 1

I suppose  you want to know the intervals in which the function is raising or decreasing...  the function is continuous and derivable .. Find the derivative f ' (x)

f (x) = 0.3  x^4 - 0.8 x^3  - 3 x^2 + 7.2 x + 11 
f '(x) =  1.2  x^3  - 2.4 x^2  - 6 x  + 7.2
        = 1.2 g(x)
where
            g(x) =  x^3 - 2 x^2  - 5 x + 6
  
    we want to find when  g(x) is 0.  ie. roots of that..  It is not easy to find the roots of a polynomial of degree 3 or more....
   
   by looking at the coefficients,  we see that  1 - 2 - 5 + 6 =0... so
       g(x) = 0  for x = 1      so  x-1  is a factor of g(x)

now ,  let   g(x) = (x - 1) (x^2 + a x - 6)
             need to find  the value of a...
      expand the RHS and compare coefficients of  x^2 or x terms...
           a x^2 -1 x^2  =  -2 x^2
             so a = -1..

   so  g(x) =   (x -1 ) (x^2 - x - 6)  = (x - 1) (x - 3) (x + 2 )
  
    So   g(x) or f '(x)  = 0   for x = 1, 3 or -2.
===========
find the sign positive or negative of  f ' (x)  for  x =  -3, -1, 0 , 2, 4...
f '(-3)  negative.
f '(-2) = 0
f '(-1) = positive 
f '(0)  = positive
f '(1) =  0
f '(2) = negative
f '(3) = 0
f '(4) = positive...

so we can now say that the given polynomial f (x) is  decreasing from - infinity to -2.  Then it increases from x =  -2 to 1.    Then it decreases from  x = 1 to 3....
then again f  increases from  x = 3  to infinity...