Let's break down the given information step-by-step and use it to sketch the graph of the function $f(x)$:

1. Domain:

The domain of $f(x)$ is $(-\infty, \infty)$, meaning the function is defined for all real numbers.

2. Behavior at infinity:

• As $x \to -\infty$, $f(x) \to -\infty$. This tells us that the function decreases without bound on the left.
• As $x \to \infty$, $f(x) \to \infty$. This tells us that the function increases without bound on the right.

3. Point of interest: $f(0) = 7$:

The function passes through the point $(0, 7)$. This is a key point on the graph.

4. Critical points:

• $f'(x) = 0$ at $x = -1$ and $x = 1$. These are possible local extrema, i.e., where the function may have a local maximum or minimum.
• $f'(x) > 0$ on $(-\infty, -1) \cup (1, \infty)$. This indicates that the function is increasing on these intervals.
• $f'(x) < 0$ on $(-1, 1)$. This indicates that the function is decreasing on this interval.

5. Concavity:

• $f''(x) < 0$ on $(-\infty, 0)$. This tells us the graph is concave down (shaped like a hill) on the left side of $x = 0$.
• $f''(x) > 0$ on $(0, \infty)$. This tells us the graph is concave up (shaped like a bowl) on the right side of $x = 0$.

Step-by-step sketch construction:

1. From left to right:

   • The function starts decreasing as $x \to -\infty$ since $\lim_{x \to -\infty} f(x) = -\infty$.
   • The function increases on $(-\infty, -1)$, so the slope is positive in this region.
   • At $x = -1$, there's a critical point, which is a local maximum since $f'(x)$ changes from positive to negative at this point.
   • The function decreases on $(-1, 1)$, with a critical point at $x = 1$, which is a local minimum because $f'(x)$ changes from negative to positive at this point.

2. At $x = 0$:

   • We know $f(0) = 7$, so the function passes through the point $(0, 7)$.
   • The function is concave down on $(-\infty, 0)$ and concave up on $(0, \infty)$, so there's an inflection point at $x = 0$.

3. To the right:

   • The function increases on $(1, \infty)$, and since $\lim_{x \to \infty} f(x) = \infty$, the function will keep increasing without bound as $x \to \infty$.

Conclusion:

• The graph starts at negative infinity for $x \to -\infty$, increases until $x = -1$, where there's a local maximum.
• It then decreases to a local minimum at $x = 1$, passing through the point $(0, 7)$, where the concavity changes.
• The graph then increases indefinitely as $x \to \infty$, while maintaining concave up behavior.

If you need an actual visual of this graph, I can generate one for you. Would you like that?

Expanding Questions:

1. How do concavity and inflection points affect the shape of a graph?
2. Can you explain how the first derivative test helps identify local extrema?
3. What is the significance of a function’s behavior at infinity?
4. How does knowing the critical points help when sketching a graph?
5. What additional information could be helpful for refining this graph sketch?

Tip:

To fully understand how a function behaves, always analyze both the first and second derivatives. The first derivative reveals increasing/decreasing behavior, while the second derivative gives insight into concavity.