Question

If tangents to the curve $y=\frac{x^4}{4}+\frac{a{x}^3}{3}+\frac{a{x}^2}{2}+x+1,xϵR$
always lie below the curve, then range of $a$ is

• A. $[0,3]$
• B. $(-\infty ,\infty )$
• C. $(-\infty ,3)$
• D. $(-\infty ,3)\cup (3,\infty )$

Answer 1

The correct option is A $[0,3]$

For tangents lie below to curve, the curve of $f\left(x\right)$ should be convex.
$\therefore f^{\prime \prime }\left(x\right)\ge 0\forall xϵR$
$f^{\prime }\left(x\right)=x^3+a{x}^2+ax+1$
$f^{\prime \prime }\left(x\right)=3x^2+2ax+a\ge 0$
So $D\le 0$
$\Rightarrow 4a^2-4\times 3a\le 0$
$\Rightarrow 0\le a\le 3$