Question

is the graph of $r(x)=-8x^{3}-7x^{2}+2x + 3$ concave up or down at the point with $x$-coordinate $-2$?
select the correct answer below:
concave down
concave up

Explanation:

Step1: Find the first - derivative

Using the power rule \((x^n)^\prime=nx^{n - 1}\), for \(r(x)=-8x^{3}-7x^{2}+2x + 3\), we have \(r^\prime(x)=-24x^{2}-14x + 2\).

Step2: Find the second - derivative

Differentiate \(r^\prime(x)\) again. Using the power rule, \(r^{\prime\prime}(x)=-48x-14\).

Step3: Evaluate the second - derivative at \(x = - 2\)

Substitute \(x=-2\) into \(r^{\prime\prime}(x)\): \(r^{\prime\prime}(-2)=-48\times(-2)-14\).
First, calculate \(-48\times(-2)=96\). Then \(r^{\prime\prime}(-2)=96 - 14=82\).

Answer:

Concave up