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TIFR ECE 2022 | Question: 6

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Consider a degree-$5$ polynomial function $f:(-\infty, \infty) \rightarrow(-\infty, \infty)$. If $f$ exhibits at least four local maxima, which of the following is necessarily true? (Note: A local maximum is a point where the function value is the maximum in a sufficiently small neighbourhood.)

  1. $f(x)>0, x \in(-\infty, \infty)$
  2. $f(50)<0$
  3. The seventh derivative of $f(x)$ is negative for some $x \in[0,100]$
  4. $f$ has exactly $4$ local maxima 
  5. None of the above

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