Question

1. the graph of y = 3x^4 - 16x^3 + 24x^2 + 48 is concave down for (a) x < 0 (b) x > 0 (c) x < -2 or x > -2/3 (d) x < 2/3 or x > 2 (e) 2/3 < x < 2 no calculator

Explanation:

Step1: Find the first - derivative

Differentiate $y = 3x^{4}-16x^{3}+24x^{2}+48$ using the power rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$.
$y'=12x^{3}-48x^{2}+48x$

Step2: Find the second - derivative

Differentiate $y'$ with the power rule.
$y'' = 36x^{2}-96x + 48$

Step3: Set $y''<0$ for concave - down

$36x^{2}-96x + 48<0$. Divide through by 12: $3x^{2}-8x + 4<0$.

Step4: Factor the quadratic

Factor $3x^{2}-8x + 4$ as $(3x - 2)(x - 2)<0$.

Step5: Find the solution of the inequality

The roots of the quadratic equation $3x^{2}-8x + 4 = 0$ are $x=\frac{2}{3}$ and $x = 2$. The solution to $(3x - 2)(x - 2)<0$ is $\frac{2}{3}

Answer:

E. $\frac{2}{3}