If the function $\mathrm{f}\left(\mathrm{x}\right) = \sqrt{2\mathrm{x} - 3}$ is invertible then find its inverse. Hence prove that $\left({\mathrm{fof}}^{- 1}\right)\left(\mathrm{x}\right) = \mathrm{x}$ 

If ${\tan }^{-1}\left(\mathrm{a}\right) +\hspace{0.17em}{\tan }^{-1}\left(\mathrm{b}\right) +\hspace{0.17em}{\tan }^{-1}\left(\mathrm{c}\right) = \mathrm{\pi }$, prove that a + b + c = abc.

Use properties of determinants to solve for x:

$\left|\begin{array}{ccc}\mathrm{x} +\mathrm{a}& \mathrm{b}& \mathrm{c}\\ \mathrm{c}& \mathrm{x} +\mathrm{b}& \mathrm{a}\\ \mathrm{a}& \mathrm{b}& \mathrm{x} +\hspace{0.17em}\mathrm{c}\end{array}\right| = 0 \mathrm{and} \mathrm{x} \ne 0$

Show that the function f(x) = $$\begin{gathered} \left\{{\mathrm{x}}^2, \mathrm{x} \le 1 \\ \frac{1}{2}, \mathrm{x} > 1\right\} \end{gathered}$$ is continuous at x = 1 but not differentiable.

15.

Verify Rolle’s theorem for the following function:

$\mathrm{f}\left(\mathrm{x}\right) = {\mathrm{e}}^{- \mathrm{x}}\sin \left(\mathrm{x}\right)$ on [0, $\mathrm{\pi }$]

If x = $\tan \left(\frac{1}{\mathrm{a}}\log \left(\mathrm{y}\right)\right)$, prove that $(1 + {\mathrm{x}}^2)\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} + (2\mathrm{x} - \mathrm{a})\frac{d\mathrm{y}}{d\mathrm{x}} = 0$

Evaluate: $\int {\tan }^{-1}\left(\sqrt{\mathrm{x}}\right)\mathrm{dx}$

Find the points on the curve 4x³  - 3x + 5 at which the equation of the tangent is parallel to the x-axis.

Solve: $\sin \left(\mathrm{x}\right)\frac{d\mathrm{y}}{d\mathrm{x}} - \mathrm{y} = \sin \left(\mathrm{x}\right).\tan \left(\frac{\mathrm{x}}{2}\right)$

The population of a town grows at the rate of 10% per year. Using differential equation, find how long will it take for the population to grow 4 times.