Question

The set of values of $a$ for which the function $f\left(x\right)=\frac{a{x}^3}{3}+(a+2)x^2+(a-1)x+2$ possesses a negative point of inflection

• A. $(-\infty ,-2)U(0,\infty )$
• B. $\{-\frac{4}{5}\}$
• C. $(-2,0)$
• D. $\varphi$

Answer 1

The correct option is A $(-\infty ,-2)U(0,\infty )$

$f^{\prime }\left(x\right)=a{x}^2+2(a+2)x+(a-1)f^{\prime \prime }\left(x\right)=2ax+2(a+2)$

$f\left(x\right)$ will have point of inflection when $f^{\prime \prime }\left(x\right)=0$

$\Rightarrow 2ax+2(a+2)=0\Rightarrow x=\frac{-(a+2)}{a}x<0\Rightarrow \frac{-(a+2)}{a}<0\Rightarrow \frac{(a+2)}{a}>0\Rightarrow a\in (-\infty ,-2)\cup (0,\infty )$

Hence, $a\in (-\infty ,-2)\cup (0,\infty )$