AP EAMCET · Maths · Application of Derivatives
Which one of the following functions is monotonically increasing in its domain?
- A \(f(x)=\log (1+x)-x+\frac{x^2}{2}\)
- B \(g(x)=2 \operatorname{Tan}^{-1} x-x-1\)
- C \(h(x)=4 \cos x+x\)
- D \(u(x)=\log (1+x)-\frac{x}{x+1}\)
Answer & Solution
Correct Answer
(A) \(f(x)=\log (1+x)-x+\frac{x^2}{2}\)
Step-by-step Solution
Detailed explanation
\(f(x)=\log (1+x)-x+\frac{x^2}{2}\) \(f'(x) = \frac{1}{1+x} - 1 + x = \frac{1-(1+x)+x(1+x)}{1+x} = \frac{1-1-x+x+x^2}{1+x} = \frac{x^2}{1+x}\) For \(x > -1\), \(1+x > 0\) and \(x^2 \ge 0\). \(f'(x) = \frac{x^2}{1+x} \ge 0\). Therefore, \(f(x)\) is monotonically increasing in its…
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