Answer

Verified

354.6k+ views

**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 .

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Onam is the main festival of which state A Karnataka class 7 social science CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Which are the Top 10 Largest Countries of the World?

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Who was the founder of muslim league A Mohmmad ali class 10 social science CBSE

Select the word that is correctly spelled a Twelveth class 10 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers