To plot the graph of the function $f(x) = 3x^5 - 5x^3 + 1$, we can analyze and understand its behavior:

1. Function Analysis:

   • Degree: The highest power of $x$ is $5$, so this is a quintic function. Its graph may exhibit up to 4 turning points.
   • Leading Coefficient: The coefficient of $x^5$ is positive, meaning the ends of the graph rise to $+\infty$ as $x \to +\infty$ and fall to $-\infty$ as $x \to -\infty$.
   • Symmetry: Since the function is a combination of odd-powered terms and a constant, $f(x)$ has no specific symmetry (neither even nor odd).
   • Intercepts: $f(0) = 1$, so the y-intercept is at $(0, 1)$.

2. Critical Points and Turning Points: To find turning points, we calculate the derivative $f'(x)$, set it to zero, and solve for $x$: $f'(x) = 15x^4 - 15x^2 = 15x^2(x^2 - 1) = 15x^2(x - 1)(x + 1)$ The critical points are $x = 0, \pm 1$.

   We'll classify these points and examine the graph behavior.

Let me plot the graph for $f(x)$. numpy as np import matplotlib.pyplot as plt

Define the function f(x)

def f(x): return 3 * x5 - 5 * x3 + 1

Generate x values and compute y values

x = np.linspace(-2, 2, 500) y = f(x)

Plot the graph

plt.figure(figsize=(8, 6)) plt.plot(x, y, label=r"$f(x) = 3x^5 - 5x^3 + 1$", color="blue") plt.axhline(0, color="black", linewidth=0.8, linestyle="--") plt.axvline(0, color="black", linewidth=0.8, linestyle="--") plt.title("Graph of $f(x) = 3x^5 - 5x^3 + 1$", fontsize=14) plt.xlabel("$x$", fontsize=12) plt.ylabel("$f(x)$", fontsize=12) plt.grid(alpha=0.3) plt.legend(fontsize=12) plt.show()Here's the graph of the function $f(x) = 3x^5 - 5x^3 + 1$.

The key features of the graph include:

• The y-intercept at $(0, 1)$.
• The general behavior dictated by the degree (quintic function with end behaviors $x \to \pm \infty$).
• Turning points around the critical points $x = -1, 0, 1$.

Would you like details about the critical points, turning points, or any specific behavior?

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Related Questions:

1. How do you find the inflection points for this function?
2. What are the intervals of increase and decrease for $f(x)$?
3. What is the significance of the derivative $f'(x)$ in analyzing graphs?
4. How would the graph change if $f(x)$ had even-powered terms only?
5. Can this function's roots be determined exactly, or are approximations necessary?

Tip: When analyzing polynomial functions, always consider their degree and leading coefficient to predict the graph's end behavior!