Q1. Find an extremal for the functional \[ f(x)=\int_{1}^{t_{f}}\left[2 x(t)+\frac{1}{2} \dot{x}^{2}(t)\right] d t \] The boundary conditions are \( x(1)=4, x\left(t_{f}\right)=4 \) and \( t_{f}>1 \) is free.

Show that \( F(x)=|x| \) is continuous everywhere.

USing first principle prove that \( y=\sin x \) then \( \frac{d y}{d x}=\cos x \)

\( \lim _ { x \rightarrow 0 } \frac { 3 ( x - \operatorname { cts } x ) } { x } \)

By using Newton-Raphson Method, determine the root for \( x^{3}+2 x^{2}-1=0 \). Given \( x_{0}=0.5 \). Give the answer correct to four decimal places. Dengan mengomenan

Suppose \( f \) is continuous on \( [1,5] \) and the only solutions of the equation \( f(x)= \) 6 are \( x=1 \) and \( x=4 \). If \( f(2)=8 \), explain why \( f(3)>6 \). Show that \( F(x)=|x| \) is continuous evervwhere.