Extrema, Concavity & Inflection - Part(1)
Calculus questions with fully worked solutions are presented below.
Applications of the first and second
derivatives
include determining intervals of increase and decrease, local maxima and minima,
concavity,
and points of inflection.
Question 1
For the function
\[
f(x) = (x + 3)(x - 2)^3
\]
find:
- a) the intervals of increase and decrease of \( f \)
- b) the values of \( x \) for which \( f \) has a local maximum or minimum
- c) the intervals of concavity and the inflection point(s)
Solution to Question 1
-
First derivative:
\[
f'(x) = (x - 2)^3 + 3(x + 3)(x - 2)^2
\]
\[
f'(x) = (x - 2)^2(4x + 7)
\]
-
Critical points are obtained from \( f'(x) = 0 \):
\[
x = 2 \quad \text{and} \quad x = -\frac{7}{4}
\]
-
The sign table of \( f'(x) \) is shown below:
| \(x\) |
\((-\infty,-\tfrac{7}{4})\) |
\(-\tfrac{7}{4}\) |
\((-\tfrac{7}{4},2)\) |
\(2\) |
\((2,+\infty)\) |
| \((x-2)^2\) |
\(+\) |
\(+\) |
\(+\) |
\(0\) |
\(+\) |
| \(4x+7\) |
\(-\) |
\(0\) |
\(+\) |
\(+\) |
\(+\) |
| \(f'(x)\) |
\(-\) |
\(0\) |
\(+\) |
\(0\) |
\(+\) |
| \(f(x)\) |
\(\searrow\) |
\(\min\) |
\(\nearrow\) |
\(\text{no ext.}\) |
\(\nearrow\) |
-
From the sign table, \( f \) is decreasing on
\[
(-\infty, -\tfrac{7}{4})
\]
and increasing on
\[
(-\tfrac{7}{4}, +\infty)
\]
-
b) Since \( f'(x) \) changes sign at \( x = -\tfrac{7}{4} \), the function has a local minimum at
\[
x = -\tfrac{7}{4}
\]
Although \( f'(2) = 0 \), there is no local extremum at \( x = 2 \) because the sign of \( f'(x) \) does not change.
-
c) Second derivative:
\[
f''(x) = 2(x - 2)(4x + 7) + 4(x - 2)^2
\]
\[
f''(x) = 6(x - 2)(2x + 1)
\]
-
The sign table of \( f''(x) \) is shown below:
| \(x\) |
\((-\infty,-\tfrac{1}{2})\) |
\(-\tfrac{1}{2}\) |
\((-\tfrac{1}{2},2)\) |
\(2\) |
\((2,+\infty)\) |
| \((x-2)\) |
\(-\) |
\(-\) |
\(-\) |
\(0\) |
\(+\) |
| \((2x+1)\) |
\(-\) |
\(0\) |
\(+\) |
\(+\) |
\(+\) |
| \(f''(x)\) |
\(+\) |
\(0\) |
\(-\) |
\(0\) |
\(+\) |
| \(f(x)\) |
\(\text{concave up}\) |
\(\text{inflection}\) |
\(\text{concave down}\) |
\(\text{inflection}\) |
\(\text{concave up}\) |
-
The graph of \( f \) is:
- concave up on \( (-\infty, -\tfrac{1}{2}) \) and \( (2, +\infty) \)
- concave down on \( (-\tfrac{1}{2}, 2) \)
Question 2
Given
\[
f(x) = e^{x^2 - x}
\]
find:
- a) the intervals of increase and decrease
- b) the value(s) of \( x \) where \( f \) has a local extremum
- c) the intervals of concavity and any inflection point(s)
Solution to Question 2
-
First derivative:
\[
f'(x) = (2x - 1)e^{x^2 - x}
\]
-
The only critical point is:
\[
x = \tfrac{1}{2}
\]
The sign table of \( f'(x) \) is shown below:
| \(x\) |
\((-\infty,\tfrac{1}{2})\) |
\(\tfrac{1}{2}\) |
\((\tfrac{1}{2},+\infty)\) |
| \(e^{x^2-x}\) |
\(+\) |
\(+\) |
\(+\) |
| \(2x-1\) |
\(-\) |
\(0\) |
\(+\) |
| \(f'(x)\) |
\(-\) |
\(0\) |
\(+\) |
| \(f(x)\) |
\(\text{decreasing}\) |
\(\min\) |
\(\text{increasing}\) |
-
The function is decreasing on \( (-\infty, \tfrac{1}{2}) \) and increasing on \( (\tfrac{1}{2}, +\infty) \).
-
b) Since \( f'(x) \) changes sign at \( x = \tfrac{1}{2} \), the function has a local minimum at
\[
x = \tfrac{1}{2}
\]
-
c) Second derivative:
\[
f''(x) = 2e^{x^2 - x} + (2x - 1)^2 e^{x^2 - x}
\]
\[
f''(x) = \big[2 + (2x - 1)^2\big] e^{x^2 - x}
\]
-
Since \( f''(x) > 0 \) for all \( x \), the graph is concave up on \( (-\infty, +\infty) \) and has no inflection point.
More Calculus Questions
All calculus questions with answers
Calculus tutorials and theory