Q. For what values of$x$, the function $f(x) = x^{4} - 4x^{3} + 4x^{2} + 40$ is monotonic decreasing?
A) $0 < x < 1$
B) $1 < x < 2$
C) $2 < x < 3$
D) $4 < x < 5$
Explanation
(b)
$f(x) = x^{4} - 4x^{3} + 4x^{2} + 40$
$f^{''}(x) = 4x^{3} - 12x^{2} + 8x$
For monoatonic decreasing $f^{''}(x) < 0$
$\Rightarrow \ \ \ \ \ \ \ \ x\left( 4x^{2} \right) - 12x + 8$<0
$\ \ \ x(x^{2} - 3x + 2) < 0$
$\Rightarrow \ \ \ \ x(x - 1)(x - 2) < 0$
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$\Rightarrow \ \ \ \ x \in ( - \infty,\ 0) \cup (1,\ 2)$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.