85297
The function \(f(x)=\frac{x}{\log x}\) increases on the interval
1 \((0, \infty)\)
2 \((0, \mathrm{e})\)
3 \((\mathrm{e}, \infty)\)
4 None of these
Explanation:
(C) : Given \(f(x)=\frac{x}{\log x}\) \(f(x)\) is defined for \(x>0\) \(f^{\prime}(x)=\frac{\log x \times 1-x \frac{1}{x}}{(\log x)^{2}} \Rightarrow f^{\prime}(x)=\frac{\log x-1}{(\log x)^{2}}\) \(\therefore \mathrm{f}^{\prime}(\mathrm{x})>0\) \(\log x>1\) \(\log _{\mathrm{e}} \mathrm{x}>\log _{\mathrm{e}} \mathrm{e}\) \(\mathrm{x}>\mathrm{e}\) \(\therefore \mathrm{x} \in(\mathrm{e}, \infty)\)
[JCECE-2011]
Application of Derivatives
85298
The function \(f(x)=\tan x-x\)
1 always increases
2 always decreases
3 never increases
4 sometimes increases and sometimes decreases
Explanation:
(A) : Given, \(\mathrm{f}(\mathrm{x})=\tan \mathrm{x}-\mathrm{x}\) On differentiating w.r.t. to \(x\) we get - \(f^{\prime}(\mathrm{x})=\sec ^{2} \mathrm{x}-1\) \(f^{\prime}(\mathrm{x})>0 \forall \mathrm{x} \in \mathrm{R}\) So, \(\mathrm{f}^{\prime}(\mathrm{x})\) is always increases.
[JCECE-2018]
Application of Derivatives
85299
The function \(f(x)=2-3 x\) is :
1 increasing
2 decreasing
3 neither decreasing nor increasing
4 none of the above
Explanation:
(B): If \(\mathrm{f}(\mathrm{x})\) is a function, it will be increasing or decreasing if \(\mathrm{f}^{\prime}(\mathrm{x})>0 \mathrm{f}^{\prime}(\mathrm{x})\lt 0\). We have, \(f(x)=2-3 x\) On differentiating w.r.t. to \(\mathrm{x}\) we get - \(\mathrm{f}^{\prime}(\mathrm{x})=-3\lt 0\) \(\therefore\) Function is decreasing for every value of \(\mathrm{x}\). Alternate solution - Let \(y=f(x)=2-3 x\) \(y+3 x=2\) \(\frac{\mathrm{x}}{2 / 3}+\frac{\mathrm{y}}{2}=1\) It is clear from the figure that for increasing the value of \(x\) from \(-\infty\) to \(\infty\), we will get the decreasing value of \(y\) from \(\infty\) to \(-\infty\). \(\therefore\) It is decreasing function.
Application of Derivatives
85300
A point is in motion along the curve \(12 y=x^{3}\). The \(x\)-coordinate, changes faster than \(y\) coordinate, if
NEET Test Series from KOTA - 10 Papers In MS WORD
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Application of Derivatives
85297
The function \(f(x)=\frac{x}{\log x}\) increases on the interval
1 \((0, \infty)\)
2 \((0, \mathrm{e})\)
3 \((\mathrm{e}, \infty)\)
4 None of these
Explanation:
(C) : Given \(f(x)=\frac{x}{\log x}\) \(f(x)\) is defined for \(x>0\) \(f^{\prime}(x)=\frac{\log x \times 1-x \frac{1}{x}}{(\log x)^{2}} \Rightarrow f^{\prime}(x)=\frac{\log x-1}{(\log x)^{2}}\) \(\therefore \mathrm{f}^{\prime}(\mathrm{x})>0\) \(\log x>1\) \(\log _{\mathrm{e}} \mathrm{x}>\log _{\mathrm{e}} \mathrm{e}\) \(\mathrm{x}>\mathrm{e}\) \(\therefore \mathrm{x} \in(\mathrm{e}, \infty)\)
[JCECE-2011]
Application of Derivatives
85298
The function \(f(x)=\tan x-x\)
1 always increases
2 always decreases
3 never increases
4 sometimes increases and sometimes decreases
Explanation:
(A) : Given, \(\mathrm{f}(\mathrm{x})=\tan \mathrm{x}-\mathrm{x}\) On differentiating w.r.t. to \(x\) we get - \(f^{\prime}(\mathrm{x})=\sec ^{2} \mathrm{x}-1\) \(f^{\prime}(\mathrm{x})>0 \forall \mathrm{x} \in \mathrm{R}\) So, \(\mathrm{f}^{\prime}(\mathrm{x})\) is always increases.
[JCECE-2018]
Application of Derivatives
85299
The function \(f(x)=2-3 x\) is :
1 increasing
2 decreasing
3 neither decreasing nor increasing
4 none of the above
Explanation:
(B): If \(\mathrm{f}(\mathrm{x})\) is a function, it will be increasing or decreasing if \(\mathrm{f}^{\prime}(\mathrm{x})>0 \mathrm{f}^{\prime}(\mathrm{x})\lt 0\). We have, \(f(x)=2-3 x\) On differentiating w.r.t. to \(\mathrm{x}\) we get - \(\mathrm{f}^{\prime}(\mathrm{x})=-3\lt 0\) \(\therefore\) Function is decreasing for every value of \(\mathrm{x}\). Alternate solution - Let \(y=f(x)=2-3 x\) \(y+3 x=2\) \(\frac{\mathrm{x}}{2 / 3}+\frac{\mathrm{y}}{2}=1\) It is clear from the figure that for increasing the value of \(x\) from \(-\infty\) to \(\infty\), we will get the decreasing value of \(y\) from \(\infty\) to \(-\infty\). \(\therefore\) It is decreasing function.
Application of Derivatives
85300
A point is in motion along the curve \(12 y=x^{3}\). The \(x\)-coordinate, changes faster than \(y\) coordinate, if
85297
The function \(f(x)=\frac{x}{\log x}\) increases on the interval
1 \((0, \infty)\)
2 \((0, \mathrm{e})\)
3 \((\mathrm{e}, \infty)\)
4 None of these
Explanation:
(C) : Given \(f(x)=\frac{x}{\log x}\) \(f(x)\) is defined for \(x>0\) \(f^{\prime}(x)=\frac{\log x \times 1-x \frac{1}{x}}{(\log x)^{2}} \Rightarrow f^{\prime}(x)=\frac{\log x-1}{(\log x)^{2}}\) \(\therefore \mathrm{f}^{\prime}(\mathrm{x})>0\) \(\log x>1\) \(\log _{\mathrm{e}} \mathrm{x}>\log _{\mathrm{e}} \mathrm{e}\) \(\mathrm{x}>\mathrm{e}\) \(\therefore \mathrm{x} \in(\mathrm{e}, \infty)\)
[JCECE-2011]
Application of Derivatives
85298
The function \(f(x)=\tan x-x\)
1 always increases
2 always decreases
3 never increases
4 sometimes increases and sometimes decreases
Explanation:
(A) : Given, \(\mathrm{f}(\mathrm{x})=\tan \mathrm{x}-\mathrm{x}\) On differentiating w.r.t. to \(x\) we get - \(f^{\prime}(\mathrm{x})=\sec ^{2} \mathrm{x}-1\) \(f^{\prime}(\mathrm{x})>0 \forall \mathrm{x} \in \mathrm{R}\) So, \(\mathrm{f}^{\prime}(\mathrm{x})\) is always increases.
[JCECE-2018]
Application of Derivatives
85299
The function \(f(x)=2-3 x\) is :
1 increasing
2 decreasing
3 neither decreasing nor increasing
4 none of the above
Explanation:
(B): If \(\mathrm{f}(\mathrm{x})\) is a function, it will be increasing or decreasing if \(\mathrm{f}^{\prime}(\mathrm{x})>0 \mathrm{f}^{\prime}(\mathrm{x})\lt 0\). We have, \(f(x)=2-3 x\) On differentiating w.r.t. to \(\mathrm{x}\) we get - \(\mathrm{f}^{\prime}(\mathrm{x})=-3\lt 0\) \(\therefore\) Function is decreasing for every value of \(\mathrm{x}\). Alternate solution - Let \(y=f(x)=2-3 x\) \(y+3 x=2\) \(\frac{\mathrm{x}}{2 / 3}+\frac{\mathrm{y}}{2}=1\) It is clear from the figure that for increasing the value of \(x\) from \(-\infty\) to \(\infty\), we will get the decreasing value of \(y\) from \(\infty\) to \(-\infty\). \(\therefore\) It is decreasing function.
Application of Derivatives
85300
A point is in motion along the curve \(12 y=x^{3}\). The \(x\)-coordinate, changes faster than \(y\) coordinate, if
85297
The function \(f(x)=\frac{x}{\log x}\) increases on the interval
1 \((0, \infty)\)
2 \((0, \mathrm{e})\)
3 \((\mathrm{e}, \infty)\)
4 None of these
Explanation:
(C) : Given \(f(x)=\frac{x}{\log x}\) \(f(x)\) is defined for \(x>0\) \(f^{\prime}(x)=\frac{\log x \times 1-x \frac{1}{x}}{(\log x)^{2}} \Rightarrow f^{\prime}(x)=\frac{\log x-1}{(\log x)^{2}}\) \(\therefore \mathrm{f}^{\prime}(\mathrm{x})>0\) \(\log x>1\) \(\log _{\mathrm{e}} \mathrm{x}>\log _{\mathrm{e}} \mathrm{e}\) \(\mathrm{x}>\mathrm{e}\) \(\therefore \mathrm{x} \in(\mathrm{e}, \infty)\)
[JCECE-2011]
Application of Derivatives
85298
The function \(f(x)=\tan x-x\)
1 always increases
2 always decreases
3 never increases
4 sometimes increases and sometimes decreases
Explanation:
(A) : Given, \(\mathrm{f}(\mathrm{x})=\tan \mathrm{x}-\mathrm{x}\) On differentiating w.r.t. to \(x\) we get - \(f^{\prime}(\mathrm{x})=\sec ^{2} \mathrm{x}-1\) \(f^{\prime}(\mathrm{x})>0 \forall \mathrm{x} \in \mathrm{R}\) So, \(\mathrm{f}^{\prime}(\mathrm{x})\) is always increases.
[JCECE-2018]
Application of Derivatives
85299
The function \(f(x)=2-3 x\) is :
1 increasing
2 decreasing
3 neither decreasing nor increasing
4 none of the above
Explanation:
(B): If \(\mathrm{f}(\mathrm{x})\) is a function, it will be increasing or decreasing if \(\mathrm{f}^{\prime}(\mathrm{x})>0 \mathrm{f}^{\prime}(\mathrm{x})\lt 0\). We have, \(f(x)=2-3 x\) On differentiating w.r.t. to \(\mathrm{x}\) we get - \(\mathrm{f}^{\prime}(\mathrm{x})=-3\lt 0\) \(\therefore\) Function is decreasing for every value of \(\mathrm{x}\). Alternate solution - Let \(y=f(x)=2-3 x\) \(y+3 x=2\) \(\frac{\mathrm{x}}{2 / 3}+\frac{\mathrm{y}}{2}=1\) It is clear from the figure that for increasing the value of \(x\) from \(-\infty\) to \(\infty\), we will get the decreasing value of \(y\) from \(\infty\) to \(-\infty\). \(\therefore\) It is decreasing function.
Application of Derivatives
85300
A point is in motion along the curve \(12 y=x^{3}\). The \(x\)-coordinate, changes faster than \(y\) coordinate, if
