badge image

Enroll For Free Now & Improve your performance.

×
User Icon
User Icon
User Icon
User Icon
User Icon

Thank you for registering.

One of our academic counsellors will contact you within 1 working day.

Please check your email for login details.
Close

MY CART (5)

Use Coupon: CART20 and get 20% off on all online Study Material

ITEM
DETAILS
MRP
DISCOUNT
FINAL PRICE
Total Price: Rs.

⇐ BackProceed to Checkout ⇒
There are no items in this cart.
Continue Shopping
Menu
Grade: Select Grade 

                        
if equation x 3 +ax+b=0 have only real roots, then prove that 4a 3 +27b 2 ≤ 0.
4 years ago

Answers : (1)

Ajay
209 Points 
							
{ x }^{ 3 }+ax+b\quad =\quad 0\quad =>\quad \quad { x }^{ 3 }+ax\quad =\quad -b\\ { x }^{ 3 }+ax+b\quad =\quad 0\quad \quad will\quad have\quad all\quad real\quad roots\quad if\quad \\ curve\quad y=\quad f(x)\quad =\quad { x }^{ 3 }+ax\quad intersects\quad line\quad y=g(x)\quad =\quad -b\quad at\quad three\quad points.\\ find\quad the\quad lacal\quad maxima\quad and\quad minima\quad of\quad f(x)\quad or\quad { f` }^{ }(x)\quad =\quad 0\\ \quad this\quad gives\quad 3{ x }^{ 2 }+a\quad =\quad 0\quad or\quad x=\pm \sqrt { \frac { -a }{ 3 } } \quadLet\quad m\quad =\quad \sqrt { \frac { -a }{ 3 } } \quad so\quad that\quad x=\pm m\quad are\quad points\quad of\quad local\quad maxima\quad and\quad minima\quad for\quad f(x)\\ if\quad g(x)\quad =\quad -b\quad intersects\quad f(x)\quad at\quad three\quad points\quad then\quad -b\quad lies\quad between\quad f(-m)\quad and\quad f(m)\\ f(m)\quad =\quad { m }^{ 3 }+am\quad =\quad { \left( \sqrt { \frac { -a }{ 3 } } \right) }^{ 2 }+a\left( \sqrt { \frac { -a }{ 3 } } \right) \quad =\quad 2{ \left( \frac { -a }{ 3 } \right) }^{ 3/2 }similarly\quad f(-m)\quad =\quad -\quad 2{ \left( \frac { -a }{ 3 } \right) }^{ 3/2 }the\quad required\quad condition\quad is\quad \\ -2{ \left( \frac { -a }{ 3 } \right) }^{ 3/2 }\quad <\quad -b\quad <\quad 2{ \left( \frac { -a }{ 3 } \right) }^{ 3/2 }\quad or\\ \left| b \right| \quad <\quad 2{ \left( \frac { -a }{ 3 } \right) }^{ 3/2 }\quad or\\ squaring\quad both\quad sides\quad withh\quad give\quad us\quad the\quad required\quad condition\\ 27{ b }^{ 2 }+4{ a }^{ 3 }<0
4 years ago
Think You Can Provide A Better Answer ?
Answer & Earn Cool Goodies

Other Related Questions on Algebra

Find the distance between (acos∅+bsin∅ , 0) and (0, asin∅-bcos∅). Answer me the Question Answer & Earn Cool Goodies
I have no question sry😕gfnftrzyckbkc yzjvlnofysohudtatzjvkn po gustcuvlbuf Answer & Earn Cool Goodies
if the cube roots of unity are 1,w,w^2then fnd the roots of the equation (x-1)^3+8=0
Factorize (X^10 + Y^10) Please solve in detailed way. Thanks.
Find the combined equation of the bisectors of the angles between the lines represented by...
View all Questions »


Course Features

  • 731 Video Lectures
  • Revision Notes
  • Previous Year Papers
  • Mind Map
  • Study Planner
  • NCERT Solutions
  • Discussion Forum
  • Test paper with Video Solution


Course Features

  • 101 Video Lectures
  • Revision Notes
  • Test paper with Video Solution
  • Mind Map
  • Study Planner
  • NCERT Solutions
  • Discussion Forum
  • Previous Year Exam Questions


Ask Experts

Have any Question? Ask Experts

Post Question

 
 
Answer ‘n’ Earn
Attractive Gift
Vouchers To Win!!! Click Here for details