﻿ Concavity and the Second Derivative Test Worksheet | Problems & Solutions

# Concavity and the Second Derivative Test Worksheet

Concavity and the Second Derivative Test Worksheet
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1.
Find the interval in which $f$ ($x$) = $x$$e$6$x$ is concave upward.
 a. ($\frac{1}{2}$, ∞) b. (- ∞, - ∞) c. (- $\frac{1}{3}$, ∞) d. (- ∞, $\frac{-1}{2}$)

#### Solution:

f(x) = xe6x

f ′(x) = 6x e6x + e6x
[First derivative.]

f ¢¢(x) = 36x e6x+ 6e6x + 6e6x = e6x(36x + 12)
[Second derivative.]

f(x) is concave upwards if f ¢¢(x) > 0

e6x (36x + 12) > 0

36x + 12 > 0
(e6x > 0.)

x > - 1 / 3 x (- 1 / 3, ∞)
[Solve the inequality.]

2.
Find the interval in which $f$($x$) = 7$x$(ln 8$x$) is concave upward.
 a. (0, ∞) b. (- 7 , 7 ) c. (- ∞, ∞) d. (- ∞, 0)

#### Solution:

f(x) = 7x(ln 8x)

f ′(x) = 7 + 7(ln 8x)
[First derivative.]

f ¢¢(x) = 7x
[Second derivative.]

f (x) is concave upward if f ¢¢(x) > 0

Þ 7x > 0

x > 0 x (0, ∞)
[Solve the inequality.]

3.
What is the interval in which $f$($x$) = $\frac{\left(ln\left(8x\right)\right)}{\left(x\right)}$ is concave upward ?
 a. (- ∞, ∞) b. (- ∞, $\frac{1}{8}$${e}^{\frac{3}{2}}$) c. (- $\frac{1}{8}$${e}^{\frac{3}{2}}$, ∞) d. ($\frac{1}{8}$${e}^{\frac{3}{2}}$, ∞)

#### Solution:

f(x) = (ln(8x)) / (x)

Domain of f(x) is (0, ∞)
[Evaluate.]

f ′(x) = (1 - ln 8x) /x2
[First derivative.]

f ¢¢(x) = [x2(-88x) - (1 - ln 8x)(2x)]x4

f ¢¢(x) =(- x - 2x + 2x ln 8x)x4

f ¢¢(x) = [x(2ln 8x - 3)]x4
[Second derivative.]

f(x) is concave upward if f ¢¢(x) > 0

[x(2ln 8x - 3)]x4 > 0

2ln 8x - 3 > 0
[x > 0.]

ln 8x > 3 / 2

x > 1 / 8e32 x (1 / 8e32, ∞)
[Solve the inequality.]

4.
What is the interval in which $f$ ($x$) = 3$x$3 + 27$x$2 + 12$x$ + 40 is concave downward?
 a. (- ∞, - 3) b. (- 3, ∞) c. (- ∞, 3) d. (- ∞, ∞)

#### Solution:

f(x) = 3x3 + 27x2 + 12x + 40
[Given.]

f ′(x) = 9x2 + 54x + 12
[First derivative.]

f ¢¢(x) = 18x + 54 = 18(x + 3)
[Second derivative.]

f(x) is concave downward if f ¢¢(x) < 0

18(x + 3) < 0

x < - 3 x (- ∞, - 3)
[Solve the inequality.]

5.
Find the interval in which $f$ ($x$) = 5$x$ - $\frac{41}{x}$ is concave downward.
 a. (- ∞, 0) b. (- ∞, 82) c. (- ∞, ∞) d. (0, ∞)

#### Solution:

f(x) = 5x - 41x
[Given function.]

f ′(x) = 5 + 41x2
[First derivative.]

f ¢¢(x) = - 82x3
[Second derivative.]

f(x) is concave downward if f ¢¢(x) < 0

- 82x3 < 0

82x3 > 0

x3 > 0 x > 0 x (0, ∞)
[Solve the inequality.]

6.
Find the interval in which $f$ ($x$) = $e$- 3$x$ is concave downward.
 a. (3, ∞) b. (- ∞, ∞) c. Not existing d. (- ∞, 9)

#### Solution:

f (x) = e- 3x
[Given.]

f ′(x) = - 3e- 3 x
[First derivative.]

f ¢¢(x) = 9e- 3x
[Second derivative.]

Since for all real values of x, 9e- 3x > 0, f (x) is concave upward for all x (- ∞, ∞).

So there is no real value of x for which f (x) is concave downward.

7.
In (0, $\frac{\pi }{2}$), $f$ ($x$) = 3$x$ cos $x$ + 36 is
 a. Concave downward b. Concave upward in(0, $\frac{\pi }{4}$) & concave downward in ($\frac{\pi }{4}$, $\frac{\pi }{2}$) c. Concave downward in(0, $\frac{\pi }{4}$) & concave upward in ($\frac{\pi }{4}$, $\frac{\pi }{2}$). d. Concave upward

#### Solution:

f(x) = 3xcosx + 36

f ′(x) = 3cos x - 3x sin x
[First derivative.]

f ¢¢(x) = - 3sin x - 3sin x - 3x cos x = - 3(x cos x + 2sin x)
[Second derivative.]

For all x (0, π2), 3(x cosx + 2sinx) > 0

- 3(x cos x + 2sin x) < 0 f ¢¢(x) < 0

So, f (x) is concave downward in (0, π2)

8.
Find the largest interval in which $f$ ($x$) = 18$x$4 + 6$x$2 + 44 is concave upward.
 a. (0, ∞) b. (- ∞, ∞) c. (- ∞, 12) d. (18, ∞)

#### Solution:

f(x) = 18x4 + 6x2 + 44
[Given function.]

f ′(x) = 72x3 + 12x
[First derivative.]

f ¢¢(x) = 216x2 + 12 = 12(18x2 + 1)
[Second derivative.]

For all the real values of x, x2 > 0 12(18x2 + 1) > 0 f ¢¢ (x) > 0

So, f(x) is concave upward for all x (- ∞, ∞)

9.
For the function $f$ ($x$) = 8$\mathrm{ax}$2 + 2$\mathrm{bx}$ + 47 where $a$ ≠ 0, which of the following is true?
 a. If $a$ < 0, $f$ ($x$) is concave upward b. If $b$ > 0, $f$ ($x$) is concave downward c. If $b$ < 0, $f$ ($x$) is concave upward d. If $a$ > 0, $f$ ($x$) is concave upward

#### Solution:

f(x) = 8ax2 + 2bx + 47 where a ≠ 0

f ′(x) = 16ax + 2b
[First derivative.]

f ¢¢(x) = 16a
[Second derivative.]

If a > 0, f ¢¢ (16x) > 0 hence f (x) is concave upward for a > 0.

10.
What is the interval in which $f$($x$) = is concave downward ?
 a. (0, ∞) b. (- ∞, 7 ) c. (- ∞, ∞) d. (7, ∞)

#### Solution:

f (x) = x - 8x - 7

f ′(x) = [(x - 7)(1) - (x - 8)(1)](x - 7)2 = 1(x - 7)2
[First derivative.]

f ¢¢(x) = - 2(x - 7)3
[Second derivative.]

f(x) is concave downward if f ¢¢(x) < 0

- 2(x - 7)3 < 0

2(x - 7)3 > 0

(x - 7) 3 > 0

x - 7 > 0 x > 7 x (7, ∞)
[Solve the inequality.]