Question

current objective
determine concavity and find the inflection points given a function
question
determine 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 ).
select the correct answer below:
concave down
concave up

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