Question

question 6
given the function ( g(x) = 6x^3 - 27x^2 - 72x ), find the first derivative, ( g(x) ).
( g(x) = 18x^2 - 54x - 72 )
notice that ( g(x) = 0 ) when ( x = 4 ), that is, ( g(4) = 0 ).
now, we want to know whether there is a local minimum or local maximum at ( x = 4 ), so we will use the second derivative test.
find the second derivative, ( g(x) ).
( g(x) = 36x - 54 )
evaluate ( g(4) ).
( g(4) = 90 )
based on the sign of this number, does this mean the graph of ( g(x) ) is concave up or concave down at ( x = 4 )?
at ( x = 4 ) the graph of ( g(x) ) is select an answer
based on the concavity of ( g(x) ) at ( x = 4 ), does this mean that there is a local minimum or local maximum at ( x = 4 )?
at ( x = 4 ) there is a local select an answer
question help: video message instructor

Explanation:

Step1: Analyze concavity sign rule

If $g''(x) > 0$, the graph is concave up.
We know $g''(4)=90$, and $90>0$.

Step2: Link concavity to extrema

For critical points where $g'(x)=0$:

• Concave up ($g''(x)>0$) means local minimum.

Answer:

At $x = 4$ the graph of $g(x)$ is concave up
At $x = 4$ there is a local minimum