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MathematicsMultiple correct+3Question 44 of 60

JEE Advanced 2014 · Paper 1, Mathematics question 44

Let a∈Ra\in\mathbb R and let f:R→Rf:\mathbb R\to\mathbb R be given by
f(x)=x5−5x+a.f(x)=x^5-5x+a.
Then

Choose one or more options
Answer and worked solution

Official answer: B, D

Turning points. f′(x)=5x4−5=0f'(x)=5x^4-5=0 at x=±1x=\pm1.

  • Local maximum f(−1)=4+af(-1)=4+a.
  • Local minimum f(1)=a−4f(1)=a-4.

Number of real roots. Three real roots need f(−1)>0>f(1)f(-1)>0>f(1), i.e. −4<a<4-4<a<4. Otherwise the quintic crosses the axis once.

  • (A) For a>4a>4 both turning values are positive: one root. False.
  • (B) True, by the same reasoning.
  • (C) For a<−4a<-4 both are negative: one root. False.
  • (D) True.

(B), (D)

Topics:Maxima and minimaTheory of equations (higher degree)
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