If $f\left(x\right)=2x+{\cot }^{-1}x+\ln \left(\sqrt{1+x^2}-x\right)$, then $f\left(x\right):$

Important Questions on Application of Derivatives

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Alpha Question Bank for Engineering: Mathematics>Chapter 28 - Application of Derivatives>Exercise-2>Q 47

Let $g\left(x\right)=2f\left(\frac{x}{2}\right)+f\left(1-x\right)$ and $f^{\text{'}\text{'}}\left(x\right)<0$ in $0\le x\le 1$, then $g\left(x\right)$

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Alpha Question Bank for Engineering: Mathematics>Chapter 28 - Application of Derivatives>Exercise-2>Q 48

Let $f\left(x\right)=x^{\frac{m}{n}}$ for $x\in R$ where $m$ and $n$ are integers, $m$ even and $n$ odd and $0<m<n$. Then

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Alpha Question Bank for Engineering: Mathematics>Chapter 28 - Application of Derivatives>Exercise-2>Q 49

Let $f$ and $g$ be two differentiable functions defined on an interval I such that $f\left(x\right)\ge 0$ and $g\left(x\right)\le 0$ for all $x\in I$ and $f$ is strictly decreasing on $I$ while $g$ is strictly increasing on $I$ then

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Alpha Question Bank for Engineering: Mathematics>Chapter 28 - Application of Derivatives>Exercise-2>Q 50

Let $\mathrm{\varphi }\left(x\right)={\left(f\left(x\right)\right)}^3-3{\left(f\left(x\right)\right)}^2+4f\left(x\right)+5x+3\sin x+4\cos x\forall x\in R$, where $f\left(x\right)$ is a differentiable function $\forall x\in R$, then

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Alpha Question Bank for Engineering: Mathematics>Chapter 28 - Application of Derivatives>Exercise-2>Q 51

If $p, q \text{and} r$ be real, then the intervals in which, $f\left(x\right)=\left|\begin{array}{ccc}x+p^2& pq& pr\\ pq& x+q^2& qr\\ pr& qr& x+r^2\end{array}\right|$.

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Alpha Question Bank for Engineering: Mathematics>Chapter 28 - Application of Derivatives>Exercise-2>Q 52

If $f\left(x\right)=\frac{x^2}{2-2\cos x};g\left(x\right)=\frac{x^2}{6x-6\sin x}$ where $0<x<1$, then

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Alpha Question Bank for Engineering: Mathematics>Chapter 28 - Application of Derivatives>Exercise-2>Q 53

Let $f\left(x\right)=\frac{x}{\sin x} ,x\in \left(0,\frac{\pi }{2}\right)$. Then the interval in which at least one root of the equation lie$\frac{2}{x-f\left(\frac{\pi }{12}\right)}+\frac{3}{x-f\left(\frac{\pi }{4}\right)}+\frac{4}{x-f\left(\frac{5\pi }{12}\right)}=0$.

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Alpha Question Bank for Engineering: Mathematics>Chapter 28 - Application of Derivatives>Exercise-2>Q 54

Let $f\left(x\right)={\left(x^2-1\right)}^n\left(x^2+x+1\right)·f\left(x\right)$ has local extremum at $x=1$ if