extreme f(x)=-2x^2-12x-16 — step-by-step solution

1) Find the strict local extrema of

- 2 x^{2} - 12 x - 16

on ℝ: the derivative must be zero and change sign

⇥1.1) Let's perform the differentiation:

\frac{d}{d x} (- 2 x^{2} - 12 x - 16)

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

\frac{d}{d x} (- 2 x^{2} - 12 x - 16)

\frac{d}{d x} (- 2 x^{2}) + \frac{d}{d x} (- 12 x) + 0

⇥

\frac{d}{d x} (- 2 x^{2}) + \frac{d}{d x} (- 12 x) + 0

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

\frac{d}{d x} (- 2 x^{2})

-2 \cdot \frac{d}{d x} (x^{2})

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

-12

■The derivative of x^n is n*x^(n-1):

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

2 x

■Adding 0 changes nothing:

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

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

⇥

- 2 \cdot 2 x - 12

⇥ⓘ Multiplications are done before additions and subtractions:

2 \cdot 2

4

⇥

- 4 x - 12

⇥The first derivative is

- 4 x - 12

⇥1.2) Let's solve in

ℝ

- 4 x - 12 = 0

⇥ⓘ We add

4 x

to both sides

⇥

-12 = 4 x

⇥ⓘ Both sides are divided by

4

⇥

x = - \frac{12}{4}

⇥ⓘ We simplify the fraction by dividing the numerator and the denominator by

4

, their greatest common divisor:

- \frac{12}{4}

-3

⇥Solution found:

x = -3

⇥1.3) Let's perform the differentiation:

\frac{d}{d x} (- 4 x - 12)

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

\frac{d}{d x} (- 4 x - 12)

\frac{d}{d x} (- 4 x) + 0

⇥

\frac{d}{d x} (- 4 x) + 0

⇥ⓘ Adding 0 changes nothing:

\frac{d}{d x} (- 4 x) + 0

\frac{d}{d x} (- 4 x)

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

\frac{d}{d x} (- 4 x)

-4

⇥

-4

⇥The second derivative is

-4

2) At

x = -3

, calculate

f^{''} (-3) = -4

x = -3

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

⇥

- 2 \cdot {(-3)}^{2} - 12 \cdot (-3) - 16

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

{(-3)}^{2}

(-3) \cdot (-3)

9

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

-12 \cdot (-3)

12 \cdot 3

(10 + 2) \cdot 3

10 \cdot 3 + 2 \cdot 3

30 + 6

36

⇥

- 2 \cdot 9 + 36 - 16

⇥ⓘ Multiplications are done before additions and subtractions:

2 \cdot 9

18

■Start from

36

and move back

16

on the number line:

36 - 16

20

⇥

- 18 + 20

⇥ⓘ Start from

-18

and move forward

20

on the number line, going past zero:

- 18 + 20

2

⇥

2

The y-coordinate is f(

-3

) =

2

▶Strict local extrema:

{(-3, 2)}