Question

The function \[f\] defined by \[f\left( x \right) = 4{x^4} - 2x + 1\] in increasing for
A.\[x < 1\]
B.\[x > 0\]
C.\[x < \dfrac{1}{2}\]
D.\[x > \dfrac{1}{2}\]

Answer 1

Hint: For the function to be increasing we take the first derivative test of the given function and then put it greater than zero , and then find the value of \[x\] for the function to be increasing . To check the function to be increasing, put the different values of \[x\] in the given function and draw the graph .

Complete step-by-step answer:
Given : \[f\left( x \right) = 4{x^4} - 2x + 1\]
Now differentiating the function for the first derivative test we have
\[{f^1}\left( x \right) = 16{x^3} - 2\] , on simplifying we get ,
\[{f^1}\left( x \right) = 2\left( {8{x^3} - 1} \right)\] .
Now putting the first derivative greater than zero , we have
\[{f^1}\left( x \right) > 0\]
On putting the values we get
\[ = 2\left( {8{x^3} - 1} \right) > 0\] , on solving we get ,
\[ = \left( {8{x^3} - 1} \right) > 0\]
On further solving we get
\[ = 8{x^3} > 1\] , on simplifying we get ,
\[ = {x^3} > \dfrac{1}{8}\]
Now taking the cube root on both sides we get ,
\[ = x > \dfrac{1}{2}\] .
Therefore , at \[x > \dfrac{1}{2}\] the function will be increasing .
Therefore , option ( D ) is the correct answer for the given question .
So, the correct answer is “Option D”.

Note: The derivative of a function is used to determine whether the function is increasing or decreasing on any intervals in its domain . If \[{f^1}\left( x \right) > 0\] at each point in an interval \[I\], then the function is said to be increasing on I. \[{f^1}\left( x \right) > 0\] at each point in an interval I, then the function is said to be decreasing on \[I\] , Because the derivative is zero or does not exist only at critical points of the function, it must be positive or negative at all other points where the function exists . In determining intervals where a function is increasing or decreasing, you first find domain values where all critical points will occur .