Find Intervals of Increase and Concavity from f' Graph

Answer

The function $f$ is increasing on $(-4, -2) \cup (1, 4)$, decreasing on $(-2, 1)$, concave down on $(-0.5, 2.5)$, and concave up on $(-4, -0.5) \cup (2.5, 4)$. Your provided answers are correct.

Explanation

The graph shown is the derivative $f'(x)$. To analyze $f(x)$, we must interpret the sign and slope of this graph.

1. Increasing and Decreasing Behavior Recall that $f(x)$ increases when the derivative $f'(x) > 0$ and decreases when $f'(x) < 0$.

   • Looking at the graph, $f'(x)$ is above the $x$-axis on $(-4, -2)$ and $(1, 4)$, meaning $f$ is increasing there.
   • $f'(x)$ is below the $x$-axis on $(-2, 1)$, so $f$ is decreasing on that interval. ⚠️ This step is required on exams: $f'(x) > 0 \implies f$ is increasing; $f'(x) < 0 \implies f$ is decreasing.

2. Concavity Analysis Concavity is determined by the sign of the second derivative $f''(x)$, which corresponds to the slope of the graph of $f'(x)$.

   • $f$ is concave down when $f''(x) < 0$, which occurs where the graph of $f'(x)$ has a negative slope (is decreasing). Observing the graph, $f'(x)$ decreases between $x = -0.5$ and $x = 2.5$.
   • $f$ is concave up when $f''(x) > 0$, which occurs where the graph of $f'(x)$ has a positive slope (is increasing). This happens on $(-4, -0.5)$ and $(2.5, 4)$. $f''(x) = \frac{d}{dx}[f'(x)]$ The second derivative is the rate of change of the first derivative.

Property of $f(x)$	Condition on $f'(x)$
Increasing	$f'(x) > 0$
Decreasing	$f'(x) < 0$
Concave Up	$f'(x)$ is increasing
Concave Down	$f'(x)$ is decreasing

Final Answer

The intervals are:

• Increasing: $\boxed{(-4, -2) \cup (1, 4)}$
• Decreasing: $\boxed{(-2, 1)}$
• Concave down: $\boxed{(-0.5, 2.5)}$
• Concave up: $\boxed{(-4, -0.5) \cup (2.5, 4)}$

Common Mistakes

• Confusing $f'$ with $f$: A common error is looking at where the graph is increasing/decreasing to determine the concavity, rather than looking at the slope of the given graph. Remember: look at the function values ($y$-axis) for monotonicity, but look at the trend (upward/downward motion) for concavity.
• Misinterpreting the axes: Students often try to find the zeros of the original function $f$ instead of the zeros of $f'$, which are the local extrema of $f$. Always verify if the graph provided is $f$ or $f'$.