Find the values of a and b such that the function f defined by
\(f(x)=\begin{cases} \frac { x-4 }{ |x-4|\quad } +a \ \ \ \ , if\quad x<4 \\ \quad a+b\quad \ \ \ \ \ \ \ , \ if\quad x=4\quad is\quad a\quad continuous\quad function\quad function\quad at\quad x=4 \\ \frac { x-4 }{ |x-4| } +b \ \ \ \ \ \ \ \ \ , \ if\quad x>4 \end{cases}\)

4

(a)

Differentiate w.r.t. x or find \(\frac { dy }{ dx } \) : \({ 2 }^{ cos^{ 2 }x }\)

4

(a)

Differentiate w.r.t. x or find \(\frac { dy }{ dx } \) : \(x={ e }^{ \theta }\left( \theta +\frac { 1 }{ \theta } \right) ,y={ e }^{ -\theta }\left( \theta -\frac { 1 }{ \theta } \right) \)

4

(a)

If y= tan^-1 x , find \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \)in terms of y alone.

4

(a)

Show that the function \(f(x)=\left\{\begin{array}{c} \frac{e^{1 / x}-1}{e^{1 / x}+1}, \text { when } x \neq 0 \\ 0, \quad \text { when } x=0 \end{array}\right.\) is discontinuous at x = 0.

4

(a)

Let \(f(x)=\left\{\begin{array}{ll} \frac{1-\cos 4 x}{x^{2}}, & \text { if } x<0 \\ a, & \text { if } x=0 \\ \frac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4} & \text { if } x>0 \end{array}\right.\) for what value of a f is continuous at x = 0 ?

4

(a)

Show that the function \(f(x)=|x-5|\) is continuous but not differentiable at x = 5.

4

(a)

If \(y=\tan ^{-1}\left[\frac{\sqrt{1+x^{2}}+\sqrt{1-x^{2}}}{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}\right],-1 then find the derivative of y.

4

(a)

Using mean value theorem, prove that there is a point on the curve y = 2x² - 5x + 3 between the points A(1, 0) and B(2, 1), where tangents are parallel to the chord. Also, find that point.

4

(a)