given the function ( g(x)=8 x^{3}-12 x^{2}-48 x ), find the first derivative, ( g^{prime}(x) ).\n(…

given the function ( g(x)=8 x^{3}-12 x^{2}-48 x ), find the first derivative, ( g^{prime}(x) ).\n( g^{prime}(x)= )\nnotice that ( g^{prime}(x)=0 ) when ( x = 2 ), that is, ( g^{prime}(2)=0 ).\nnow, we want to know whether there is a local minimum or local maximum at ( x = 2 ), so we will use the second derivative test.\nfind the second derivative, ( g^{prime prime}(x) ).\n( g^{prime prime}(x)= )\nevaluate ( g^{prime prime}(2) ).\n( g^{prime prime}(2)= )\nbased on the sign of this number, does this mean the graph of ( g(x) ) is concave up or concave down at ( x = 2 )?\nanswer either up or down -- watch your spelling!!\nat ( x = 2 ) the graph of ( g(x) ) is concave \nbased on the concavity of ( g(x) ) at ( x = 2 ), does this mean that there is a local minimum or local maximum at ( x = 2 )?\nanswer either minimum or maximum -- watch your spelling!!\nat ( x = 2 ) there is a local

given the function ( g(x)=8 x^{3}-12 x^{2}-48 x ), find the first derivative, ( g^{prime}(x) ).\n( g^{prime}(x)= )\nnotice that ( g^{prime}(x)=0 ) when ( x = 2 ), that is, ( g^{prime}(2)=0 ).\nnow, we want to know whether there is a local minimum or local maximum at ( x = 2 ), so we will use the second derivative test.\nfind the second derivative, ( g^{prime prime}(x) ).\n( g^{prime prime}(x)= )\nevaluate ( g^{prime prime}(2) ).\n( g^{prime prime}(2)= )\nbased on the sign of this number, does this mean the graph of ( g(x) ) is concave up or concave down at ( x = 2 )?\nanswer either up or down -- watch your spelling!!\nat ( x = 2 ) the graph of ( g(x) ) is concave \nbased on the concavity of ( g(x) ) at ( x = 2 ), does this mean that there is a local minimum or local maximum at ( x = 2 )?\nanswer either minimum or maximum -- watch your spelling!!\nat ( x = 2 ) there is a local

Answer

Explanation:

Step1: Find the first derivative

Use the power rule ((x^n)^\prime = nx^{n - 1}). For (g(x)=8x^{3}-12x^{2}-48x), (g^\prime(x)=8\times3x^{2}-12\times2x - 48) (g^\prime(x)=24x^{2}-24x - 48)

Step2: Find the second derivative

Differentiate (g^\prime(x)=24x^{2}-24x - 48) using the power rule. (g^{\prime\prime}(x)=24\times2x-24) (g^{\prime\prime}(x)=48x - 24)

Step3: Evaluate (g^{\prime\prime}(2))

Substitute (x = 2) into (g^{\prime\prime}(x)). (g^{\prime\prime}(2)=48\times2-24) (g^{\prime\prime}(2)=96 - 24=72)

Step4: Determine concavity

Since (g^{\prime\prime}(2)=72>0), by the second - derivative test for concavity, the graph of (y = g(x)) is concave up at (x = 2).

Step5: Determine local extrema

By the second - derivative test for local extrema: if (g^\prime(c)=0) and (g^{\prime\prime}(c)>0), then (g(x)) has a local minimum at (x = c). Here (c = 2), (g^\prime(2) = 0) and (g^{\prime\prime}(2)>0), so (g(x)) has a local minimum at (x = 2).

Answer:

(g^\prime(x)=24x^{2}-24x - 48) (g^{\prime\prime}(x)=48x - 24) (g^{\prime\prime}(2)=72) At (x = 2) the graph of (g(x)) is concave up At (x = 2) there is a local minimum