Let $f$ be the function defined by $f(x) = \frac{3}{2x^2 - 7x + 5}$. (a) Find the slope of the line tangent to the graph of $f$ at $x = 3$. (b) Find the $x$-coordinate of each critical point of $f$ in the interval $1 < x < 2.5$. Classify each critical point as the location of a relative minimum, a relative maximum, or neither. Justify your answers. (c) Using the identity that $\frac{3}{2x^2 - 7x + 5} = \frac{2}{2x - 5} - \frac{1}{x - 1}$, evaluate $\int_{5}^{\infty} f(x)\, dx$ or show that the integral diverges. (d) Determine whether the series $\sum_{n=5}^{\infty} \frac{3}{2n^2 - 7n + 5}$ converges or diverges. State the conditions of the test used for determining convergence or divergence.
Let $f$ be the function defined by $f(x) = \frac{3}{2x^2 - 7x + 5}$.\\
(a) Find the slope of the line tangent to the graph of $f$ at $x = 3$.\\
(b) Find the $x$-coordinate of each critical point of $f$ in the interval $1 < x < 2.5$. Classify each critical point as the location of a relative minimum, a relative maximum, or neither. Justify your answers.\\
(c) Using the identity that $\frac{3}{2x^2 - 7x + 5} = \frac{2}{2x - 5} - \frac{1}{x - 1}$, evaluate $\int_{5}^{\infty} f(x)\, dx$ or show that the integral diverges.\\
(d) Determine whether the series $\sum_{n=5}^{\infty} \frac{3}{2n^2 - 7n + 5}$ converges or diverges. State the conditions of the test used for determining convergence or divergence.