Question

The roots of (x – 41)⁴⁹ + (x – 49)⁴¹ + (x – 2009)²⁰⁰⁹ = 0 are - 

(A) all necessarily real 

(B) non-real except one positive real root 

(C) non-real except three positive real roots 

(D) non-real except for three real roots of which exactly one is positive  

Answer 1

Correct Option :- (B) non-real except one positive real root

Explanation :-

(x – 41)⁴⁹ + (x – 49)⁴¹ + (x – 2009)²⁰⁰⁹ = 0 

f(x) = (x – 41)⁴⁹ + (x – 49)⁴¹ + (x – 2009)²⁰⁰⁹ 

f ′ (x) = 49(x – 41)⁴⁸ + 41(x – 49)⁴⁰ + 2009 (x – 2009)⁴⁸ > 0 

hence f(x) will cut x-axis only once.

1 real root .