To analyze the function $f(x) = |x| + x^2 + x^4$, let's consider its behavior across different domains of $x$.

1. Analyzing the Function on Different Domains

The absolute value function $|x|$ affects the behavior of $f(x)$ depending on whether $x$ is non-negative or negative:

• For $x \geq 0$: $|x| = x$ Thus, the function simplifies to: $f(x) = x + x^2 + x^4$

• For $x < 0$: $|x| = -x$ Hence, the function becomes: $f(x) = -x + x^2 + x^4$

2. Behavior of $f(x)$ Based on the Sign of $x$

Case 1: $x \geq 0$

When $x \geq 0$, the function is: $f(x) = x + x^2 + x^4$ All the terms $x, x^2, x^4$ are non-negative for $x \geq 0$, so $f(x) \geq 0$. This function is continuous and differentiable for all $x \geq 0$.

Case 2: $x < 0$

When $x < 0$, the function is: $f(x) = -x + x^2 + x^4$ Since $x < 0$, $-x > 0$. The terms $x^2$ and $x^4$ are also positive, meaning that $f(x) > 0$ for $x < 0$ as well. Thus, $f(x)$ remains continuous and differentiable for all $x$.

3. Key Characteristics of the Function $f(x) = |x| + x^2 + x^4$

• Continuity: The function $f(x)$ is continuous everywhere because $|x|, x^2,$ and $x^4$ are continuous functions.
• Differentiability: $f(x)$ is differentiable everywhere. Though the absolute value function $|x|$ has a non-differentiable point at $x = 0$, the combination $f(x) = |x| + x^2 + x^4$ remains differentiable at $x = 0$ due to the smoothness added by the $x^2$ and $x^4$ terms.

4. Graph and Monotonicity

• For $x \geq 0$: $f(x) = x + x^2 + x^4$ is strictly increasing since each term $x, x^2, x^4$ is increasing.

• For $x < 0$: $f(x) = -x + x^2 + x^4$. Even though $-x$ is positive and decreasing, $x^2 + x^4$ increases faster as $x$ moves away from zero. Hence, $f(x)$ also increases.

5. Conclusion

The function $f(x) = |x| + x^2 + x^4$ is:

• Continuous and differentiable for all $x$.
• Strictly increasing for all $x$ due to the dominating terms $x^2$ and $x^4$.

Summary

$f(x) = |x| + x^2 + x^4$ is a smooth, strictly increasing function that is positive for all real numbers.

Would you like to see the detailed graph of this function or have any specific questions about its properties?

Additional Questions:

1. How does the graph of $f(x) = |x| + x^2 + x^4$ look for different ranges of $x$?
2. What are the derivatives of $f(x) = |x| + x^2 + x^4$ for $x \geq 0$ and $x < 0$?
3. How would the function change if one of the terms were removed, such as $x^4$?
4. Can we determine the concavity of $f(x)$ over its domain?
5. What is the limit of $f(x)$ as $x$ approaches infinity or negative infinity?

Tip: When analyzing functions involving absolute values, consider breaking down the function into cases for positive and negative $x$ to understand its behavior fully.