Answer :

Given:- Function f(x)= {x (x – 2)}2


Theorem:- Let f be a differentiable real function defined on an open interval (a,b).


(i) If f’(x) > 0 for all , then f(x) is increasing on (a, b)


(ii) If f’(x) < 0 for all , then f(x) is decreasing on (a, b)


Algorithm:-


(i) Obtain the function and put it equal to f(x)


(ii) Find f’(x)


(iii) Put f’(x) > 0 and solve this inequation.


For the value of x obtained in (ii) f(x) is increasing and for remaining points in its domain it is decreasing.


Here we have,


f(x)= {x (x – 2)}2


f(x)= {[x2–2x]}2



f’(x)= 2(x2–2x)(2x–2)


f’(x)= 4x(x–2)(x–1)


For f(x) lets find critical point, we must have


f’(x) = 0


4x(x–2)(x–1)= 0


x(x–2)(x–1)= 0


x =0, 1, 2





Now, lets check values of f(x) between different ranges


Here points x = 0, 1, 2 divide the number line into disjoint intervals namely, (–, 0),(0, 1), (1, 2) and (2, ∞)


Lets consider interval (–, 0) and (1, 2)


In this case, we have x(x–2)(x–1)< 0


Therefore, f’(x) < 0 when x < 0 and 1< x < 2


Thus, f(x) is strictly decreasing on interval (–∞, 0)(1, 2)


Now, consider interval (0, 1) and (2, ∞)


In this case, we have x(x–2)(x–1)> 0


Therefore, f’(x) > 0 when 0< x < 1 and x < 2


Thus, f(x) is strictly increases on interval (0, 1)(2, ∞)



Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
caricature
view all courses
RELATED QUESTIONS :

Find the intervalRD Sharma - Volume 1

Find the intervalRD Sharma - Volume 1

Find the intervalRD Sharma - Volume 1

Find the intervalRD Sharma - Volume 1

Find the intervalRD Sharma - Volume 1

Find the intervalRD Sharma - Volume 1

Show that the altMathematics - Board Papers

Find the intervalRD Sharma - Volume 1

Find the intervalRD Sharma - Volume 1

Find the intervalRD Sharma - Volume 1