numpy as np import matplotlib.pyplot as plt

Defining the function f(x) = sqrt(2x^2 + 5)

def f(x): return np.sqrt(2*x**2 + 5)

Generating x values for plotting

x = np.linspace(-10, 10, 400) y = f(x)

Plotting the graph of f(x)

plt.figure(figsize=(8,6)) plt.plot(x, y, label=r'$f(x) = \sqrt{2x^2 + 5}$', color='blue')

Marking the extremum (if any) and asymptote

As the function has no extrema in the real domain, we'll focus on asymptotic behavior.

plt.axhline(0, color='black',linewidth=1) # x-axis plt.axvline(0, color='black',linewidth=1) # y-axis

Asymptote behavior at infinity (no real asymptote but we'll mention the general behavior)

plt.text(8, 10, r'$\lim_{x \to \infty} f(x) = \infty$', fontsize=12, color='red')

Labeling the axes

plt.xlabel('x') plt.ylabel('f(x)') plt.title('Graph of f(x) = sqrt(2x^2 + 5)')

Adding grid

plt.grid(True)

Show legend

plt.legend()

Save the graph as a PDF file

plt.savefig("/mnt/data/graph_f_x_sqrt_2x2_plus_5.pdf", format='pdf')

Display the graph

plt.show()Here is the graph of the function $f(x) = \sqrt{2x^2 + 5}$.

Key Observations:

• Extremum: The function does not have any extremum points (like minima or maxima) because the function is monotonically increasing for all real $x$.
• Asymptote: There is no vertical asymptote, as the function is defined for all real $x$. However, the function grows indefinitely as $|x| \to \infty$, which can be seen from the graph. As $x \to \infty$ or $x \to -\infty$, the function increases without bound.

The graph has been saved as a PDF. You can download it from the link below:

Download Graph PDF

Let me know if you need further details or clarifications.

Related Questions:

1. How do the horizontal and vertical asymptotes behave for similar functions like $\sqrt{3x^2 + 2}$?
2. What happens if we modify the function by changing the constant in the square root, e.g., $f(x) = \sqrt{2x^2 + 10}$?
3. How do you find the domain and range of functions like $f(x) = \sqrt{2x^2 + 5}$?
4. How can you graph other square root functions and analyze their asymptotic behavior?
5. What are the practical applications of square root functions in real-world problems?

Tip: When analyzing functions of the form $f(x) = \sqrt{ax^2 + b}$, notice that the function behaves similarly to a line for large $|x|$, and the growth rate can be approximated by $f(x) \approx \sqrt{a} |x|$ for large values of $x$.