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GATE 2021 EE – Question 36

Engineering Mathematics · Calculus: Mean value theorems, Theorems of integral calculus, Evaluation of definite and improper integrals, Partial derivatives, Multiple integrals, Fourier series · 2 marks · Multiple choice

In the open interval $(0,1)$, the polynomial $p(x)=x^4-4x^3+2$ has

  1. two real roots
  2. one real root
  3. three real roots
  4. no real roots

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Show answer and explanation

Correct answer: (B) one real root

Explanation

$p(0)=2>0$ and $p(1)=1-4+2=-1<0$, so there is a root in $(0,1)$. $p'(x)=4x^2(x-3)<0$ on $(0,1)$, so $p$ is strictly decreasing there and the root is unique.