Question

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

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

Answer 1

Answer: (A)

$(-\infty ,-2)\cup (0,\infty )$

Given, $f\left(x\right)=\frac{a{x}^3}{3}+(a+2)x^2+(a-1)x+2$

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

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

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

$\Rightarrow x=-\frac{a+2}{a}$

For negative point of inflection $\frac{a+2}{a}>0$

$\Rightarrow aϵ(-\infty ,-2)\cup (0,\infty )$