Let's analyze the given function $f(x) = \frac{1}{10}x^4 - x^2 - 2$. Here are some steps to better understand this function:

1. Key Features of the Function

• Degree: This is a fourth-degree polynomial, so its graph will exhibit behavior typical of quartic functions, including the possibility of multiple turning points.
• Leading Coefficient: The term $\frac{1}{10}x^4$ has a positive coefficient, so the graph will eventually rise as $x \to \pm\infty$.
• Symmetry: Since all terms contain only even powers of $x$, the function is symmetric about the $y$-axis, meaning it is an even function.

2. Critical Points and Extrema

To find critical points, calculate the derivative and solve $f'(x) = 0$: $f'(x) = \frac{4}{10}x^3 - 2x = \frac{2}{5}x^3 - 2x$ Factorize: $f'(x) = \frac{2}{5}x(x^2 - 5)$ Setting $f'(x) = 0$, we find: $x = 0 \quad \text{or} \quad x = \pm\sqrt{5}$ These are the critical points.

3. Second Derivative Test

To determine whether the critical points correspond to maxima, minima, or points of inflection, compute the second derivative: $f''(x) = \frac{6}{5}x^2 - 2$ Evaluate $f''(x)$ at the critical points:

• At $x = 0$: $f''(0) = -2$ (local maximum at $x = 0$).
• At $x = \pm\sqrt{5}$: $f''(\pm\sqrt{5}) = \frac{6}{5}(5) - 2 = 6 - 2 = 4 > 0$ (local minima at $x = \pm\sqrt{5}$).

4. Intercepts

• $y$-Intercept: Set $x = 0$: $f(0) = -2$ So the $y$-intercept is $(0, -2)$.
• $x$-Intercepts: Solve $f(x) = 0$: $\frac{1}{10}x^4 - x^2 - 2 = 0$ Multiply through by 10: $x^4 - 10x^2 - 20 = 0$ Let $u = x^2$: $u^2 - 10u - 20 = 0$ Solve using the quadratic formula: $u = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(-20)}}{2(1)} = \frac{10 \pm \sqrt{100 + 80}}{2} = \frac{10 \pm \sqrt{180}}{2} = \frac{10 \pm 6\sqrt{5}}{2} = 5 \pm 3\sqrt{5}$ Since $u = x^2$, we have: $x^2 = 5 + 3\sqrt{5} \quad \text{and} \quad x^2 = 5 - 3\sqrt{5}$ Taking square roots gives four $x$-intercepts: $x = \pm\sqrt{5 + 3\sqrt{5}}, \quad x = \pm\sqrt{5 - 3\sqrt{5}}$

5. End Behavior

As $x \to \pm\infty$, the $\frac{1}{10}x^4$ term dominates, so $f(x) \to +\infty$.

Would you like me to graph the function or go deeper into any of these steps? 😊

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5 Related Questions:

1. How do quartic functions behave differently from cubic functions?
2. What are the implications of a function being even or odd?
3. Can you explain the quadratic substitution method used here in detail?
4. How do we confirm the symmetry of a function mathematically?
5. How does the second derivative test help classify critical points?

Tip:

Always check the behavior of higher-order terms when analyzing polynomials, as they dictate the end behavior of the graph.