Determine the monotonic intervals of this function y=2x^3-6x^2-18x-7.

targetepd 2022-08-13 Answered
Determine the monotonic intervals of the functions
i need to determine the monotonic intervals of this function y = 2 x 3 6 x 2 18 x 7. I tried the below but i am not sure if i am doing it right.
My work: y = 2 x 3 6 x 2 18 x 7 6 x 2 12 x 18 = 0 6 ( x 2 2 x 3 ) = 0 ( x 3 ) ( x + 1 ) x 3 = 0 x + 1 = 0 x = 3 , x = 1
so my function increases when x [ 3 , + [ and decreases when x [ 1 , 3 ] ] , 1 ].
Please i want to know how to solve this problem any help with explanation will be appreciated. thanks in advanced
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Answers (1)

Favatc6
Answered 2022-08-14 Author has 17 answers
Step 1
You have correctly found the derivative d y d x = 6 x 2 12 x 18 = 6 ( x 2 2 x 3 ) = 6 ( x 3 ) ( x + 1 ) and where it is zero, but you have not quite got the intervals correct.
The derivatives is positive if and only if ( x 3 ) ( x + 1 ) > 0 which is positive if and only if ( x > 3  and  x > 1 )  or  ( x < 3  and  x < 1 ) i.e. if and only if ( x > 3 )  or  ( x < 1 ).
Step 3
So the function is (strictly) increasing on the interval ( , 1 ] and on the interval [ 3 , ).
The function is (strictly) decreasing on the interval [ 1 , 3 ]
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