Given the graph of the first derivative f'(x), how do you find where the original function f(x) is increasing?

The function f(x) is increasing wherever its first derivative f'(x) is strictly positive (above the x-axis).

Given the graph of the second derivative f''(x), how do you find the inflection points of f(x)?

Inflection points of f(x) occur exactly where the second derivative f''(x) crosses the x-axis, changing its sign from positive to negative or negative to positive.

How do you match a function to its derivative graphs?

Identify key features: wherever the original function has a local maximum or minimum, its first derivative must cross the x-axis (equal zero). Wherever the original function changes concavity, the second derivative must cross the x-axis.

How do you find concavity from the graph of a first derivative?

The concavity of the original function f(x) corresponds to the slope of the first derivative f'(x). When the graph of f'(x) is increasing, f(x) is concave up. When the graph of f'(x) is decreasing, f(x) is concave down.

Calculus Graph Analysis: First & Second Derivatives Practice

Question 1

The graphs of a function $f$, its first derivative $f'(x)$, and its second derivative $f''(x)$ are shown below. Identify which graph represents $f$, $f'(x)$, and $f''(x)$.

Graphs of a function and its first and second derivatives

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Solution:

Graph (I) is entirely above the x-axis ($y>0$). Therefore, it cannot represent a derivative of graphs (II) or (III), since both show intervals of increase and decrease, which require sign changes in the derivative. Hence, graph (I) represents the function $f$.
Graph (I) has a maximum at $x=0$. Thus, $f'(0)=0$. Among the remaining graphs, only graph (III) crosses the x-axis at $x=0$. Therefore, graph (III) represents $f'(x)$.
Graph (II) must represent $f''(x)$. It is negative near $x=0$, indicating that $f$ is concave down at that point, which matches the shape of graph (I) around $x=0$.

Question 2

The graph of the first derivative $f'(x)$ of a function $f$ is shown below.

a) For what values of $x$ is $f$ increasing?
b) For what values of $x$ is $f$ decreasing?
c) At which values of $x$ does $f$ have a local maximum or minimum?
d) Where is the graph of $f$ concave up? Concave down?
e) Where are the points of inflection of $f$?

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Solution:

a) Function $f$ is increasing where $f'(x)>0$: $$(-\infty,-6.6)\cup(0,3.6)$$
b) Function $f$ is decreasing where $f'(x)<0$: $$(-6.6,0)\cup(3.6,+\infty)$$
c) Local maxima occur where $f'(x)=0$ and changes from positive to negative: $$x=-6.6\quad\text{and}\quad x=3.6$$ Local minima occur where $f'(x)=0$ and changes from negative to positive: $$x=0$$
d) Concavity is determined by the sign of $f''(x)$ (the slope of $f'(x)$).
- $f'(x)$ is increasing on $(-4,2)$, so $f''(x)>0$ and $f$ is concave up on: $$(-4,2)$$
- $f'(x)$ is decreasing on $(-\infty,-4)\cup(2,+\infty)$, so $f''(x)<0$ and $f$ is concave down on these intervals.
e) Points of inflection occur where $f''(x)=0$ and changes sign (where $f'(x)$ has local extrema). From the graph, these occur at: $$x=-4\quad\text{and}\quad x=2$$

Question 3

The graph of the second derivative $f''(x)$ of a function $f$ is shown below.

a) Where does $f'(x)$ have a local maximum or minimum?
b) Where is $f$ concave up?
c) Where is $f$ concave down?
d) Where are the points of inflection of $f$?

Show Solution

Solution:

a) $f'(x)$ has a local maximum where $f''(x)=0$ and changes from positive to negative: $$x=1$$ $f'(x)$ has local minima where $f''(x)=0$ and changes from negative to positive: $$x=-2\quad\text{and}\quad x=3$$
b) $f$ is concave up where $f''(x)>0$: $$(-2,1)\cup(3,+\infty)$$
c) $f$ is concave down where $f''(x)<0$: $$(-\infty,-2)\cup(1,3)$$
d) Points of inflection occur where $f''(x)$ changes sign (crosses the x-axis): $$x=-2,\;1,\;3$$

First and Second Derivatives Questions with Answers (Part 2)

Question 1

Question 2

Question 3

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