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.

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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...*