Lyons Vargas

08/14/2024 · Middle School

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 \( n+3=0 \).