is the graph of $r(x)=-8x^{3}-7x^{2}+2x + 3$ concave up or down at the point with $x$-coordinate…

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

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

Answer

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