extreme f(t)=t^3-12t+2 — step-by-step solution

1) Find the strict local extrema of

t^{3} - 12 t + 2

on ℝ: the derivative must be zero and change sign

⇥1.1) Let's differentiate:

\frac{d}{d t} (t^{3} - 12 t + 2)

⇥ⓘ The derivative of a sum is the sum of the derivatives:

\frac{d}{d t} (t^{3} - 12 t + 2)

\frac{d}{d t} (t^{3}) + \frac{d}{d t} (- 12 t) + 0

⇥

\frac{d}{d t} (t^{3}) + \frac{d}{d t} (- 12 t) + 0

⇥ⓘ The derivative of x^n is n*x^(n-1):

\frac{d}{d t} (t^{3})

3 t^{2}

■Adding 0 changes nothing:

\frac{d}{d t} (- 12 t) + 0

\frac{d}{d t} (- 12 t)

■A constant factor stays in front of the derivative, (k*u)' = k*u':

\frac{d}{d t} (- 12 t)

-12

⇥

3 t^{2} - 12

⇥The first derivative is

3 t^{2} - 12

⇥1.2) Let's solve in

ℝ

3 t^{2} - 12 = 0

⇥ⓘ Both sides are divided by

3

⇥

t^{2} = 4

⇥The two numbers whose square is

4

are

-2

and

2

⇥Solutions found:

t = {-2, 2}

⇥1.3) Let's perform the differentiation:

\frac{d}{d t} (3 t^{2} - 12)

⇥ⓘ The derivative of a sum is the sum of the derivatives:

\frac{d}{d t} (3 t^{2} - 12)

\frac{d}{d t} (3 t^{2}) + 0

⇥

\frac{d}{d t} (3 t^{2}) + 0

⇥ⓘ Adding 0 changes nothing:

\frac{d}{d t} (3 t^{2}) + 0

\frac{d}{d t} (3 t^{2})

■A constant factor stays in front of the derivative, (k*u)' = k*u':

\frac{d}{d t} (3 t^{2})

3 \cdot \frac{d}{d t} (t^{2})

■The derivative of x^n is n*x^(n-1):

\frac{d}{d t} (t^{2})

2 t

⇥

3 \cdot 2 t

⇥ⅰ

3 \cdot 2

6

⇥

6 t

⇥The second derivative is

6 t

2) At

t = -2

, calculate

f^{''} (-2) = 6 \cdot (-2) = -12

t = -2

, the second derivative is negative: this is a strict local maximum

⇥

{(-2)}^{3} - 12 \cdot (-2) + 2

⇥ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

{(-2)}^{3}

(-2) \cdot (-2) \cdot (-2)

-8

■Multiplications are done before additions and subtractions. Two minus signs cancel out, the product of two negative numbers is positive:

-12 \cdot (-2)

12 \cdot 2

(10 + 2) \cdot 2

10 \cdot 2 + 2 \cdot 2

20 + 4

24

⇥

- 8 + 24 + 2

⇥ⓘ Start from

-8

and move forward

24

on the number line, going past zero:

- 8 + 24

16

⇥

16 + 2

⇥ⓘ Start from

16

and move forward

2

on the number line:

16 + 2

18

⇥

18

The y-coordinate is f(

-2

) =

18

3) At

t = 2

, calculate

f^{''} (2) = 6 \cdot 2 = 12

t = 2

, the second derivative is positive: this is a strict local minimum

⇥

2^{3} - 12 \cdot 2 + 2

⇥ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

2^{3}

2 \cdot 2 \cdot 2

8

■Multiplications are done before additions and subtractions:

12 \cdot 2

(10 + 2) \cdot 2

10 \cdot 2 + 2 \cdot 2

20 + 4

24

⇥

8 - 24 + 2

⇥ⓘ Start from

8

and move back

24

on the number line, going below zero:

8 - 24

-16

⇥

- 16 + 2

⇥ⓘ Start from

-16

and move forward

2

on the number line, getting closer to zero:

- 16 + 2

-14

⇥

-14

The y-coordinate is f(

2

) =

-14

This odd-degree polynomial is unbounded above and below on ℝ: it has no global extremum

▶Strict local extrema:

{(-2, 18), (2, -14)}