Question

Find the intervals in which the following function is (a) increasing (b) decreasing f(x)=2x3−9x2+12x+30

Answer 1

Answer:

• f(x) is increasing on ( -∞, 1 ) and on ( 2, ∞ )
• f(x) is decreasing on ( 1, 2 )

Hello.  I hope this helps you.

Step-by-step explanation:

The function is increasing/decreasing according as its derivative is positive/negative.  So we need the derivative.

f' (x) = 6x² - 18x + 12 = 6 ( x² - 3x + 2 ) = 6 ( x - 1 ) ( x - 2)

The derivative is equal to zero at x=1 and x=2.  These are the places where the sign of the derivative could change between positive and negative.

These places x=1 and x=2 divide the real numbers into 3 intervals:

( -∞, 1 ),  ( 1, 2 )  and  ( 2, ∞ ).

The sign of f' (x) is the same throughout each of these, so just check some convenient point in each interval to get the signs.

• f'(0) = 12 is positive, so f'(x) is positive throughout the interval ( -∞, 1 ).
• f'(1.5) = 6 ( 1.5 - 1 ) ( 1.5 - 2 ) = pos × pos × neg = negative, so f'(x) is negative throughout the interval ( 1, 2 ).
• f(3) = 6 ( 3 - 1 ) ( 3 - 2 ) = pos × pos × pos = pos, so f'(x) is positive throughout the interval ( 2, ∞ ).

Relating back to the original function f(x), It follows that

• f(x) is increasing on ( -∞, 1 ) and on ( 2, ∞ )
• f(x) is decreasing on ( 1, 2 )