extreme x^2+2x+3 — step-by-step solution

1) Find the strict local extrema of

x^{2} + 2 x + 3

on ℝ: the derivative must be zero and change sign

⇥1.1) Let's differentiate:

\frac{d}{d x} (x^{2} + 2 x + 3)

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

\frac{d}{d x} (x^{2} + 2 x + 3)

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

⇥

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

⇥ⓘ 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} (2 x) + 0

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

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

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

2

⇥

2 x + 2

⇥The first derivative is

2 x + 2

⇥1.2) Let's solve in

ℝ

2 x + 2 = 0

⇥ⓘ We subtract

2

from both sides

⇥

2 x = -2

⇥ⓘ Both sides are divided by

2

⇥

x = - \frac{2}{2}

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

2

, their greatest common divisor:

- \frac{2}{2}

-1

⇥Solution found:

x = -1

⇥1.3) Let's differentiate:

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

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

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

\frac{d}{d x} (2 x) + 0

⇥

\frac{d}{d x} (2 x) + 0

⇥ⓘ Adding 0 changes nothing:

\frac{d}{d x} (2 x) + 0

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

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

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

2

⇥

2

⇥The second derivative is

2

2) At

x = -1

, calculate

f^{''} (-1) = 2

x = -1

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

⇥

{(-1)}^{2} + 2 \cdot (-1) + 3

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

{(-1)}^{2}

(-1) \cdot (-1)

1

■Multiplications are done before additions and subtractions. A single minus sign does not cancel out, the product of a negative number by a positive number is negative:

2 \cdot (-1)

-2

⇥

1 - 2 + 3

⇥ⓘ Start from

1

and move back

2

on the number line, going below zero:

1 - 2

-1

⇥

- 1 + 3

⇥ⓘ Start from

-1

and move forward

3

on the number line, going past zero:

- 1 + 3

2

⇥

2

The y-coordinate is f(

-1

) =

2

▶Strict local extrema:

{(-1, 2)}