F(x)= -9 +4x -x^3 Use interval notation to

Question

find where f(x) is increasing and decreasing?

Answer 1

f(x) = -9 +4x -x³ 

The critical points of the function can be found by equating its first derivative to  0.

f '(x) = 4 - 3x² = 0 ⇒ x² = 4/3

                ⇒ x = ± 2/√3 

So the critical points are - 2/√3 and  2/√3 .

Testing all intervals to the left and right of these values for f '(x) = 4 - 3x², 

Now consider (-∞ , - 2/√3 ) , let us take a test point -2 in the interval .

f '(x) = 4 - 3(-2)² = 4 - 12 = -8 < 0

So f '(x) < 0 on (-∞ , - 2/√3 ) 

Now consider ( - 2/√3 , 2/√3 ) , let us take a test point 0 in the interval .

f '(x) = 4 - 3(0)² = 4 = 4 > 0

So f '(x) > 0 on ( - 2/√3 , 2/√3 ) 

Now consider ( 2/√3 , ∞ ) , let us take a test point 2 in the interval .

f '(x) = 4 - 3(2)² = 4 - 12 = -8 < 0

So f '(x) < 0 on ( 2/√3 , ∞ ) 

Hence, f(x) is increasing on ( - 2/√3 , 2/√3 )  and decreasing on (-∞ , - 2/√3 )  and ( 2/√3 , ∞ ) .