Question

Select the true statement about the graph of f(x) = -2x^3 + 6X^2 + 18X -18.

Answer 1

The y-value of the maximum must be higher that of the minimum so it is either C or D.
We now substitute the value of x of the minimum and maximum in each option to see if it matches the given value of y.
f(-1) = -2(-1)³ + 6(-1)² + 18(-1) - 18
f(-1) = -28

Thus, option C is correct.

Answer 2

f ( x ) = - 2 x³ + 6 x² + 18 x - 18
f ` ( x ) = - 6 x² + 12 x + 18 = - 6 ( x² - 2 x - 3 ) =
= - 6 ( x² - 3 x + x - 3 ) =
= - 6 ( x ( x - 3 ) + ( x - 3 ) ) =
= - 6 ( x - 3 ) ( x + 1 )
f ` ( x ) = 0  when:  x = - 1 ,  x = 3
f `` ( x ) = - 12 x + 12
f `` ( - 1 ) = 12 + 12 = 24 > 0  ( min )
f `` ( 3 ) = - 36 + 12 = - 24 < 0 ( max )
f ( - 1 ) = 2 + 6 - 18 - 18 = - 28
f ( 3 ) = - 54 + 54 + 54 - 18 = 36
Answer:
C ) ( - 1 , - 28 ) is a relative minimum and ( 3, 36 ) is a relative maximum.