Question

If $y=x^3-a{x}^2+48x+7$ is an increasing function for all real values of $x$, then $a$ lies in:

• A. $(-14,14)$
• B. $(-12,12)$
• C. $(-16,16)$
• D. $(-21,21)$

Answer 1

Answer: (B)

$(-12,12)$

$y=x^3-a{x}^2+48x+7$

$y^{\prime }=3x^2-2ax+48$

For given function to increasing for all $x$,

Discriminant of above quadratic $<0$

$\Rightarrow (-2a{)}^2-4\cdot 3\cdot 48<0$

$\Rightarrow a^2-144<0$

$\Rightarrow (a+12)(a-12)<0$

$\Rightarrow a\in (-12,12)$