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Most Important Selected Qs for JEE AdvancedMathematicsApplication of Derivatives

Let f(x) be a continuous function defined for every real x R . For any real numbers ' a ' and ' b ' that satisfy a f(b) . Then which of the followings is/are correct?

Options

  1. A_ h 0 f(2+h)-f(2) h exists and negative.
  2. BThere is always only one real root of f(x)=0
  3. CThere is always only one real root of f(x)=f(-x+1)
  4. DThere is no real root of f(x)=f(x+1)

Correct answer

D. There is no real root of f(x)=f(x+1)

Step-by-step solution

(a) f(x)=-(x-2)^ 1 / 3 _ h 0 f(2+h)-f(2) h does not exist. (b) f(x)=e^ -x solution for f(x)=0 does not exist. (c) f(x) is monotonously decreasing and f(-x+1) monotonously increases. And since its clear that y=f(x) and y=f(-x+1) meets at point ( 1 2 , f ( 1 2 ) ) , they must meet at only one point. Thus, the solution for f(x)=f(-x+1) is only one, x= 1 2 .

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