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GATE 2020 MT – Question 41

Engineering Mathematics · Calculus and vector calculus · 2 marks · Multiple choice

At x = 0.5, $f(x)=x\ln x+(1-x)\ln(1-x)+3x(1-x)$ has

  1. A local minimum
  2. A local maximum
  3. An inflection point
  4. A nonzero slope

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Correct answer: (B) A local maximum

Explanation

The derivative is $f\prime=\ln[x/(1-x)]+3-6x$, which is zero at 0.5. The second derivative is $1/x+1/(1-x)-6$, giving 4−6 = −2 at 0.5. Negative curvature identifies a local maximum, B.

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