Question

For what values of $x,$ function $f\left(x\right)=x^4-4x^3+4x^2+40$is monotonic decreasing?

• A. $0<x<1$
• B. $1<x<2$
• C. $2<x<3$
• D. $4<x<5$

Answer 1

The correct option is B

$1<x<2$

The explanation for the correct options:

Find the values of $x$:

Given the function $f\left(x\right)=x^4-4x^3+4x^2+40$

Now, differentiate with respect to $x$,

$\Rightarrow f\text{'}\left(x\right)=4x^3-12x^2+8x$

Since the function is monotonic decreasing. so, $f\text{'}\left(x\right)<0$

$\Rightarrow 4x^3–12x^2+8x<0$

$\Rightarrow x^3–3x^2+2x<0$

$\Rightarrow x(x^2–3x+2)<0$

$\Rightarrow x(x–1)(x–2)<0$

$\Rightarrow x\in (-\infty ,0)\cup (1,2)$

Thus, For $x\in (-\infty ,0)\cup (1,2)$,$f\left(x\right)=x^4-4x^3+4x^2+40$is monotonic decreasing.

Hence, Option(B) is the correct answer.