current objective\ndetermine concavity and find the inflection points given a function\nquestion\ndetermine…

current objective\ndetermine concavity and find the inflection points given a function\nquestion\ndetermine whether the graph of ( q ( x ) = - 6 x ^ { 3 } - 7 x ^ { 2 } + 8 x + 2 ) is concave up or down at the point with an ( x ) - coordinate of ( - 2 ).\nselect the correct answer below:\nconcave down\nconcave up

current objective\ndetermine concavity and find the inflection points given a function\nquestion\ndetermine whether the graph of ( q ( x ) = - 6 x ^ { 3 } - 7 x ^ { 2 } + 8 x + 2 ) is concave up or down at the point with an ( x ) - coordinate of ( - 2 ).\nselect the correct answer below:\nconcave down\nconcave up

Answer

Explanation:

Step1: Find the second - derivative of (q(x))

Use the power rule ((x^n)^\prime=nx^{n - 1}). First, find the first - derivative: (q^\prime(x)=\frac{d}{dx}(-6x^{3}-7x^{2}+8x + 2)=-18x^{2}-14x + 8). Then, find the second - derivative: (q^{\prime\prime}(x)=\frac{d}{dx}(-18x^{2}-14x + 8)=-36x-14).

Step2: Evaluate the second - derivative at (x = - 2)

Substitute (x=-2) into (q^{\prime\prime}(x)): (q^{\prime\prime}(-2)=-36\times(-2)-14). (q^{\prime\prime}(-2)=72-14). (q^{\prime\prime}(-2)=58).

Answer:

Concave up