Answer the following questions: (5 \( \times \mathbf{5}=\mathbf{2 5}) \) 1. Using mean value theorem show that, \[ \boldsymbol{x}<\log \frac{\mathbf{1}}{\mathbf{1 + x}}<\frac{\boldsymbol{x}}{\mathbf{1 + x}} \text {, if } \boldsymbol{x}<\mathbf{0}<\mathbf{1} . \] 2. Prove that the maximum value of \( x+\frac{1}{x} \) is less than its minimum value. 3. Discuss the convergence of the series \( \mathbf{1}+\frac{\mathbf{2}^{2}}{\mathbf{2 !}}+\frac{\mathbf{3}^{3}}{3!}+\frac{\mathbf{4}^{4}}{4!}+\cdots \) 4. Find eigen values and eigen vectors of the matrix \( \left(\begin{array}{ll}5 & 1 \\ 4 & 2\end{array}\right) \) 5. Prove that \( r^{n} \vec{r} \) is an irrotational for any value of \( n \) but is solenoidal if \( n+3=0 \).

Slcule os reguintes lemites: 1) \( \lim _{x \rightarrow+\infty} \frac{\ln (x)}{2 x} \) \( = \)

(b) \( \int \frac{1}{x^{2} \sqrt[5]{x^{2}}} d x= \)

A marshy region used for agricultural drainage has become contaminated with a particular element. It has been determined that flushing the area with clean water will reduce the element for a while, but it will then begin to build up again. A biologist has found that the percent of the element in the soil \( x \) months after the flushing begins is given by the function below. When will the element be reduced to a minimum? What is the minimum percent? \( f(x)=\frac{x^{2}+16}{2 x}, 1 \leq x \leq 12 \) Find \( f^{\prime}(x) \). \( f^{\prime}(x)=\frac{1}{2}-\frac{8}{x^{2}} \) When will the element be reduced to a minimum? The element will be reduced to a minimum after 4 month(s). (Type an integer or a decimal.) What is the minimum percent of the element?

3. \( \int \sec ^{2}(-5 x) d x= \)

\( 7 y ^ { \prime \prime } - 2 \sqrt { 7 } y ^ { \prime } + y = 125 \cos ( \sqrt { 7 } x ) \)