Question

The set of values of $b$ for which $f\left(x\right)=2x^4+b{x}^3+3x^2,$ has a point of inflection

• A. $b\in (-\infty ,-4)\cup (4,\infty )$
• B. $b\in (-\infty ,-4]\cup (4,\infty )$
• C. $b\in (-\infty ,-4)\cup [4,\infty )$
• D. $[-4,4]$

Answer 1

The correct option is A $b\in (-\infty ,-4)\cup (4,\infty )$

Given curve : $f\left(x\right)=2x^4+b{x}^3+3x^2$
$\Rightarrow f^{\prime \prime }\left(x\right)=24x^2+6bx+6\cdots \left(i\right)$
for point of inflection at $x=\alpha$ : $f^{\prime \prime }\left({\alpha }^{+}\right)\cdot f^{\prime \prime }\left({\alpha }^{-}\right)<0$
only possible if $D>0$ for $\left(i\right)$
$\Rightarrow 36b^2-4\cdot 24\cdot 6>0$
$\Rightarrow b^2>16$
$\Rightarrow b\in (-\infty ,-4)\cup (4,\infty )$