Find the absolute extrem of the following function on the given closed interval
f(x) = x² -12x + 10; [1, 2]

Show that the value in the conclusion of the mean value theorem for
f(x) = Ax² + Bx + c on any interval [a, b] is \(\frac{a+b}{2}\)

Write down the Taylor series expansion, of the function log x about x =1 upto three nonzero terms for x > 0.

Expand sin x in ascending powers x - \(\frac{\pi}{4}\) upto three non-zero terms.

Evaluate : \(\underset{x\rightarrow 1^{-}}{lim}(\frac{log(1-x)}{cot(\pi x)})\).

Evaluate the following limit, if necessary use l ’Hôpital Rule
\(\underset { x\rightarrow \infty }{ lim } { e }^{ -x }\sqrt { x } \)

Find the intervals of monotonicity and hence find the local extrema for the function \(f(x)=x^{\frac{2}{3}}\).

Determine the intervals of concavity of the curve y = 3+ sin x .

Find the asymptotes of the function f(x) = \(\frac{1}{x}\)

Find the slant (oblique) asymptote for the function \(f(x)=\frac { { x }^{ 2 }-6x+7 }{ x+5 } \)