First and Second Derivatives Questions with Answers (Part 2)
Calculus questions with detailed solutions are presented below.
The graphical behavior of a function \( f \), its first derivative \( f'(x) \), and its second derivative \( f''(x) \) is analyzed.
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 \( f''(x) \).
Solution to Question 1
-
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 \( f \)?
Solution to Question 2
-
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 \( f''(x) \).
-
\( f'(x) \) 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 on these intervals.
-
e)
Points of inflection occur where \( f''(x) = 0 \) and changes sign.
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 \( f \)?
Solution to Question 3
-
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:
\[
x = -2,\; 1,\; 3
\]
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