Question

The value(s) of $a,$ for which the curve $f\left(x\right)=x^4+3a{x}^3+6x^2+5$ is not situated below any of its tangent lines is/are

• A. $-\frac{2}{3}$
• B. $-\frac{1}{3}$
• C. $\frac{3}{4}$
• D. $\frac{4}{3}$

Answer 1

Answer: (A), (B), (C)

The correct options are
A $-\frac{2}{3}$
B $-\frac{1}{3}$
C $\frac{3}{4}$

$f\left(x\right)=x^4+3a{x}^3+6x^2+5$
Since, the given curve is not situated below any of its tangent lines, hence the curve must be concave.
So, $\frac{d^{2}y}{d{x}^2}>0$

$f^{\prime }\left(x\right)=4x^3+9a{x}^2+12x$
$f^{\prime \prime }\left(x\right)=12x^2+18ax+12>0$
$\Rightarrow 2x^2+3ax+2>0$
$\therefore D<0$
$\Rightarrow 9a^2-4\times 2\times 2<0$
$\Rightarrow a^2-\frac{16}{9}<0$
$\Rightarrow |a|<\frac{4}{3}$
$\Rightarrow \frac{-4}{3}<a<\frac{4}{3}$