Let f (x) = ax5 + bx4 + cx3 + dx2 + ex, Where a, b, c, d, e Ïµ R And f (x) = 0 Has A Positive Root α, Then 

Why Kaysons ?

Video lectures

Access over 500+ hours of video lectures 24*7, covering complete syllabus for JEE preparation.

Online Support

Practice over 30000+ questions starting from basic level to JEE advance level.

Live Doubt Clearing Session

Ask your doubts live everyday Join our live doubt clearing session conducted by our experts.

National Mock Tests

Give tests to analyze your progress and evaluate where you stand in terms of your JEE preparation.

Organized Learning

Proper planning to complete syllabus is the key to get a decent rank in JEE.

Test Series/Daily assignments

Give tests to analyze your progress and evaluate where you stand in terms of your JEE preparation.



Let f (x) = ax5 + bx4 + cx3 + dx2 + ex, where abcde Ïµ R and f (x) = 0 has a positive root α, then 


Correct option is

All of the above

It is given that α is a positive root of f (x) and by inspection, we have f (0) = 0  

∴ x = 0 and x = α are roots of f (x) = 0.  

By Rolle’s theorem, f’(x) = 0 has a root α1 between 0 and α i.e., 0 < α1 < α  

∴ (a) is correct.  

Clearly, f’(x) = 0 is a fourth degree equation in x and imaginary roots always occurs in pairs.  

Since x = α1 is a root of f’(x) = 0   

∴                                f’(x) = 0 will have another real root, α2 (say)   

∴ Now, α1 and α2 are real roots of f’(x) = 0  

∴ By Roll’s Theorem f’’ (x) = 0 will have a real root between α1 and α2.    

∴ (b) is correct.  

We have seen that x = 0, x = α are two real roots of f (x) = 0. As

f (x) = 0 is fifth degree equation, it will have at least three real roots. Consequently by Rolle’s  

Theorem f’(x) = 0 will have at least two real roots.  



If abc be non-zero real numbers such that  


Then the equation ax2 + bx + c = 0 will have



Find c of the Lagrange’s mean value theorem for which


Let f (x) and g (x) be differentiable for 0 ≤ x ≤ 2 such that (0) = 2, g(0) = 1 and f (2) = 8. Let there exists a real number c in [0, 2] such that f’(c) = 3g’(c) then the value of g(2) must be:


If f (x) = loge x and g(x) = x2 and c Ïµ (4, 5), then  is equal to: 


In [0, 1] lagrange’s mean value theorem is not applicable to


Let f (x) satisfy the requirement of lagrange’s mean value theorem in [0, 2]. If f (0) and   



Let f : [2, 7] and [0, ) be a continuous and differentiable function.

Then the value of (f (7) – f (2))  is (where c Ïµ (2, 7))


The equation sin x + x cos x = 0 has at least one root in the interval


Between any two real roots of the equation ex sin x – 1 = 0, the equation excos x + 1 = 0 has