Test for a Minimum

Mathematics · FE Reference Handbook section

Mathematics

2 formulas

10 exam-style examples

~49 min

All Mathematics lectures

Core formulas for this FE topic

Definitions, applicability, units, assumptions and worked examples for each relation.

Worked exam-style examples

The four ways this section is written on the real exam — thoughts first, then equations, then substitution.

Example 1

Maximum of a cost function

Total cost of a haul road is C(x) = x³ − 9x² + 24x + 5 ($1,000) with x the haul distance in km. Locate the local maximum and its value.

Given

$C(x) = x^{3} - 9x^{2} + 24x + 5$

Find

x at the local maximum and C at that point

Start with the thinking

Critical points come from C′(x) = 0; the second derivative classifies them.
A local maximum needs C″ < 0.

Figure 1 — schematic for Maximum of a cost function

Step-by-step solution

First derivative
$C'(x) = 3x^{2} - 18x + 24$
Set to zero
$3x^{2} - 18x + 24 = 0, or x^{2} - 6x + 8 = 0$
Factor
$(x - 2)(x - 4) = 0, so x = 2 and x = 4$
Second derivative
$C″(x) = 6x - 18$
Classify
$C″(2) = -6 < 0 (maximum); C″(4) = +6 > 0 (minimum)$
Value — C(2) = 8 − 36 + 48 + 5 = 25.0 ($1,000)

Answer:

Local maximum at x = 2 km, C = $25,000

Why the other options are there

x = 4 km (minimum reported)
C = 21 (arithmetic slip on 8 − 36 + 48)

Reference: FE Reference Handbook — Mathematics — Differential calculus

Example 2

Derivative evaluated at a point — Test for a Minimum

For f(x) = 5x³ − 7x² + 7x − 5, what is f ′(1)?

Given

$f(x) = 5x^{3} - 7x^{2} + 7x - 5$
$x = 1$

Find

f ′(1)

Start with the thinking

Differentiate term by term with the power rule.
Substitute only after differentiating.

Step-by-step solution

Differentiate
$f '(x) = 15x^{2} - 14x + 7$
Substituting
$f '(1) = 15(1) - 14(1) + 7$
Evaluate
$f '(1) = 15 - 14 + 7 = 8$

Answer:

f '(1) = 8

Why the other options are there

0 (f evaluated, not f ′)
13 (constant differentiated incorrectly)

Reference: FE Reference Handbook — Mathematics → Test for a Minimum

Example 3

Derivative evaluated at a point — Test for a Minimum (2)

For f(x) = 6x³ − 5x² + 1x − 5, what is f ′(1)?

Given

$f(x) = 6x^{3} - 5x^{2} + 1x - 5$
$x = 1$

Find

f ′(1)

Start with the thinking

Differentiate term by term with the power rule.
Substitute only after differentiating.

Step-by-step solution

Differentiate
$f '(x) = 18x^{2} - 10x + 1$
Substituting
$f '(1) = 18(1) - 10(1) + 1$
Evaluate
$f '(1) = 18 - 10 + 1 = 9$

Answer:

f '(1) = 9

Why the other options are there

-3 (f evaluated, not f ′)
14 (constant differentiated incorrectly)

Reference: FE Reference Handbook — Mathematics → Test for a Minimum

Example 4

Derivative evaluated at a point — Test for a Minimum (3)

For f(x) = 5x³ − 9x² + 2x − 5, what is f ′(4)?

Given

$f(x) = 5x^{3} - 9x^{2} + 2x - 5$
$x = 4$

Find

f ′(4)

Start with the thinking

Differentiate term by term with the power rule.
Substitute only after differentiating.

Step-by-step solution

Differentiate
$f '(x) = 15x^{2} - 18x + 2$
Substituting
$f '(4) = 15(16) - 18(4) + 2$
Evaluate
$f '(4) = 240 - 72 + 2 = 170$

Answer:

f '(4) = 170

Why the other options are there

179 (f evaluated, not f ′)
175 (constant differentiated incorrectly)

Reference: FE Reference Handbook — Mathematics → Test for a Minimum

Example 5

Derivative evaluated at a point — Test for a Minimum (4)

For f(x) = 2x³ − 5x² + 3x − 5, what is f ′(4)?

Given

$f(x) = 2x^{3} - 5x^{2} + 3x - 5$
$x = 4$

Find

f ′(4)

Start with the thinking

Differentiate term by term with the power rule.
Substitute only after differentiating.

Step-by-step solution

Differentiate
$f '(x) = 6x^{2} - 10x + 3$
Substituting
$f '(4) = 6(16) - 10(4) + 3$
Evaluate
$f '(4) = 96 - 40 + 3 = 59$

Answer:

f '(4) = 59

Why the other options are there

55 (f evaluated, not f ′)
64 (constant differentiated incorrectly)

Reference: FE Reference Handbook — Mathematics → Test for a Minimum

Example 6

Derivative evaluated at a point — Test for a Minimum (5)

For f(x) = 6x³ − 7x² + 2x − 5, what is f ′(3)?

Given

$f(x) = 6x^{3} - 7x^{2} + 2x - 5$
$x = 3$

Find

f ′(3)

Start with the thinking

Differentiate term by term with the power rule.
Substitute only after differentiating.

Step-by-step solution

Differentiate
$f '(x) = 18x^{2} - 14x + 2$
Substituting
$f '(3) = 18(9) - 14(3) + 2$
Evaluate
$f '(3) = 162 - 42 + 2 = 122$

Answer:

f '(3) = 122

Why the other options are there

100 (f evaluated, not f ′)
127 (constant differentiated incorrectly)

Reference: FE Reference Handbook — Mathematics → Test for a Minimum

Example 7

Derivative evaluated at a point — Test for a Minimum (6)

For f(x) = 2x³ − 8x² + 7x − 5, what is f ′(3)?

Given

$f(x) = 2x^{3} - 8x^{2} + 7x - 5$
$x = 3$

Find

f ′(3)

Start with the thinking

Differentiate term by term with the power rule.
Substitute only after differentiating.

Step-by-step solution

Differentiate
$f '(x) = 6x^{2} - 16x + 7$
Substituting
$f '(3) = 6(9) - 16(3) + 7$
Evaluate
$f '(3) = 54 - 48 + 7 = 13$

Answer:

f '(3) = 13

Why the other options are there

-2 (f evaluated, not f ′)
18 (constant differentiated incorrectly)

Reference: FE Reference Handbook — Mathematics → Test for a Minimum

Example 8

Derivative evaluated at a point — Test for a Minimum (7)

For f(x) = 4x³ − 6x² + 6x − 5, what is f ′(3)?

Given

$f(x) = 4x^{3} - 6x^{2} + 6x - 5$
$x = 3$

Find

f ′(3)

Start with the thinking

Differentiate term by term with the power rule.
Substitute only after differentiating.

Step-by-step solution

Differentiate
$f '(x) = 12x^{2} - 12x + 6$
Substituting
$f '(3) = 12(9) - 12(3) + 6$
Evaluate
$f '(3) = 108 - 36 + 6 = 78$

Answer:

f '(3) = 78

Why the other options are there

67 (f evaluated, not f ′)
83 (constant differentiated incorrectly)

Reference: FE Reference Handbook — Mathematics → Test for a Minimum

Example 9

Derivative evaluated at a point — Test for a Minimum (8)

For f(x) = 6x³ − 6x² + 1x − 5, what is f ′(2)?

Given

$f(x) = 6x^{3} - 6x^{2} + 1x - 5$
$x = 2$

Find

f ′(2)

Start with the thinking

Differentiate term by term with the power rule.
Substitute only after differentiating.

Step-by-step solution

Differentiate
$f '(x) = 18x^{2} - 12x + 1$
Substituting
$f '(2) = 18(4) - 12(2) + 1$
Evaluate
$f '(2) = 72 - 24 + 1 = 49$

Answer:

f '(2) = 49

Why the other options are there

21 (f evaluated, not f ′)
54 (constant differentiated incorrectly)

Reference: FE Reference Handbook — Mathematics → Test for a Minimum

Example 10

Derivative evaluated at a point — Test for a Minimum (9)

For f(x) = 5x³ − 9x² + 1x − 5, what is f ′(3)?

Given

$f(x) = 5x^{3} - 9x^{2} + 1x - 5$
$x = 3$

Find

f ′(3)

Start with the thinking

Differentiate term by term with the power rule.
Substitute only after differentiating.

Step-by-step solution

Differentiate
$f '(x) = 15x^{2} - 18x + 1$
Substituting
$f '(3) = 15(9) - 18(3) + 1$
Evaluate
$f '(3) = 135 - 54 + 1 = 82$

Answer:

f '(3) = 82

Why the other options are there

52 (f evaluated, not f ′)
87 (constant differentiated incorrectly)

Reference: FE Reference Handbook — Mathematics → Test for a Minimum