Question

For what value of $x$, function $f\left( x \right) = {x^4} - 4{x^3} + 4{x^2} + 40$ is monotonic decreasing?
A. $0 < x < 1$
B. $1 < x < 2$
C. $2 < x < 3$
D. $4 < x < 5$

Answer 1

Hint: In this question, we need to find the value of $x$for which the given function is monotonic decreasing. For this, we have to follow the condition such as $f'\left( x \right) < 0$. After further simplification, we will be able to find the value of $x$.

Complete step by step solution:
We know that $f\left( x \right) = {x^4} - 4{x^3} + 4{x^2} + 40$
First, we will find the derivative of the given function.
So, differentiate $f\left( x \right)$with respect to the variable$x$.
Thus, we get
$f'\left( x \right) = 4{x^3} - 12{x^2} + 8x$
Now, follow the condition $f'\left( x \right) < 0$.
$4{x^3} - 12{x^2} + 8x < 0$
Let us take the variable $4x$ common from the above expression.
$
  4x\left( {{x^2} - 3x + 2} \right) < 0 \\
  x\left( {{x^2} - 3x + 2} \right) < 0 \\
 $
By factoring, we get
$x\left( {x - 1} \right)\left( {x - 2} \right) < 0$
$x = 0,2,1$
Thus, we can say that
$f'\left( x \right) > 0$ if $0 < x < 1$ and $x > 2$
$f'\left( x \right) < 0$ if $1 < x < 2$
Therefore, for $1 < x < 2$,the function $f\left( x \right) = {x^4} - 4{x^3} + 4{x^2} + 40$ is monotonic decreasing.

Option ‘B’ is correct

Additional Information: Monotonic functions are those functions that increase or decrease over their whole domain. A function is monotonic if the sign of its first derivative does not vary. Also, a decreasing function is defined as a function that reduces in value as the independent variable grows over a specified span. Thus, a monotonically declining function lowers as $x$ rises for all real $x$. These ideas are especially suitable for examining exponential and logarithmic functions.

Note: We need to follow the proper condition of derivative for monotonic decreasing function. If the derivative of a function is less than zero for all values of the specified variable in any region for a decreasing function, the function is defined as a monotonically falling or strictly decreasing function.