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 is

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

Answer 1

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

We have,
$f\left(x\right)=\frac{a{x}^3}{3}+(a+2)x^2+(a-1)x+2$
$\Rightarrow f^{\prime }\left(x\right)=a{x}^2+2(a+2)x+(a-1)$
$\Rightarrow f^{\prime \prime }\left(x\right)=2ax+2(a+2)=0\Rightarrow x=-\frac{a+2}{a}$
For negative point of inflection
$-\frac{a+2}{a}<0$
$\Rightarrow a\in (-\infty ,-2)\cup (0,\infty )$
$f^{\prime \prime \prime }\left(x\right)=2a\ne 0$ in the above interval.