KVPY SX SB Exam 2012 Question Paper

Question : 10 of 120

Marks: +1, -0

Let n be a natural number and let a be a real number. The number of zeros of

x^{2n+1}-(2n+1)x+a=0

in the interval [-1, 1] is

Solution:

f(x)=x^{2n+1}-(2n+1)x+a=0

f'(x)=(2n+1)x^{2n}-(2n+1)

=(2n+1)(x^{2n}-1) ≤ 0

when

x ∈ [−1,1]

f(x)is strictly decreasing in [ 1,1]
f(x) cut x axis at most one point in given interval

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