find the intervals where the function is concave up and concave down. f(x)=x³ + 5x² + 6x - 1 a. concave up…

Explanation:

Step1: Find the first - derivative

Differentiate $f(x)=x^{3}+5x^{2}+6x - 1$ using the power rule. $f^\prime(x)=3x^{2}+10x + 6$.

Step2: Find the second - derivative

Differentiate $f^\prime(x)$ using the power rule. $f^{\prime\prime}(x)=6x + 10$.

Step3: Find the inflection point

Set $f^{\prime\prime}(x)=0$. So, $6x+10 = 0$, which gives $x=-\frac{5}{3}$.

Step4: Test the intervals

Choose a test - point in the interval $(-\infty,-\frac{5}{3})$, say $x=-2$. Then $f^{\prime\prime}(-2)=6\times(-2)+10=-2<0$, so the function is concave down on $(-\infty,-\frac{5}{3})$. Choose a test - point in the interval $(-\frac{5}{3},\infty)$, say $x = 0$. Then $f^{\prime\prime}(0)=6\times0 + 10=10>0$, so the function is concave up on $(-\frac{5}{3},\infty)$.

Answer:

D. Concave down on $(-\infty,-\frac{5}{3})$ Concave up on $(-\frac{5}{3},\infty)$