Question

Find the intervals in which the function f given by f ( x ) = 2 x 3 − 3 x 2 − 36 x + 7 is (a) strictly increasing (b) strictly decreasing

Answer 1

The given function is f( x )=2 x 3 −3 x 2 −36x+7

Differentiate the function with respect to x.

f ′ ( x )=6 x 2 −6x−36 =6( x 2 −x−6 ) =6( x+2 )( x−3 )

Substitute f ′ ( x )= 0 to obtain the point of maxima or minima.

f ′ ( x )=0 6( x+2 )( x−3 )=0 x=−2 or 3

Now, the points 2 and -3 divide the real line in three different intervals given by,

(a)

In the interval ( −∞,−2 ) and ( 3,∞ ) ,

f ′ ( x )>0 6 x 2 −6x−36>0

Thus, f( x ) is strictly increasing in the interval ( −∞,−2 ) and ( 3,∞ ).

(b)

In the interval ( −2,3 ),

f ′ ( x )<0 6 x 2 −6x−36<0

Thus, f( x ) is strictly decreasing in the interval ( −2,3 ).