Question

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

Solution

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.  

SIMILAR QUESTIONS

Q1

If abc be non-zero real numbers such that  

                                                                      

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

Q2

 

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

Q3

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:

Q4

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

Q6

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

Q7

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

 

Q8

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))

Q9

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

Q10

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