$\text { Find the relative extremum of } f ( x ) = - 9 x ^ { 2 } + 54 x + 2 \text { by applying the First }$ Derivative Test.
A) relative minimum: $( - 4 , - 358 )$
B) relative maximum: $( - 3 , - 241 )$
C) relative minimum: $( 4,74 )$
D) relative minimum: $( 2,74 )$
E) relative maximum: $( 3,83 )$

Question

$\text { Find the relative extremum of } f ( x ) = - 9 x ^ { 2 } + 54 x + 2 \text { by applying the First }$ Derivative Test. 
A) relative minimum: $( - 4 , - 358 )$
B) relative maximum: $( - 3 , - 241 )$
C) relative minimum: $( 4,74 )$
D) relative minimum: $( 2,74 )$
E) relative maximum: $( 3,83 )$

Quizplus · Accepted Answer

To find the relative extremum, we first find the derivative of $f(x) = -9x^2 + 54x + 2$, which is $f'(x) = -18x + 54$. Setting $f'(x) = 0$ gives $x = 3$. To determine if it's a maximum or minimum, we can check the sign of $f''(x)$ or note that the coefficient of $x^2$ in $f(x)$ is negative, indicating a concave down parabola and thus a maximum at $x = 3$. Substituting $x = 3$ into $f(x)$ gives $f(3) = -9(3)^2 + 54(3) + 2 = 83$, so the relative maximum is at $(3, 83)$.

Find the relative extremum of f(x)=−9x2+54x+2 by applying the First \text { Find the relative extremum of } f ( x ) = - 9 x ^ { 2 } + 54 x + 2 \text { by applying the First } Find the relative extremum of f(x)=−9x2+54x+2 by applying the First

$\text { Find the relative extremum of } f ( x ) = - 9 x ^ { 2 } + 54 x + 2 \text { by applying the First }$