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 - 2 x

⇥1.2) Let's solve in

ℝ

2 - 2 x = 0

⇥ⓘ We add

2 x

to both sides

⇥

2 = 2 x

⇥ⓘ 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 - 2 x)

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

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

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

⇥

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

⇥ⓘ Adding 0 changes nothing:

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

\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 negative: this is a strict local maximum

⇥

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

⇥

- 1 + 2 + 3

⇥ⓘ Start from

-1

and move forward

2

on the number line, going past zero:

- 1 + 2

1

⇥

1 + 3

⇥ⓘ Start from

1

and move forward

3

on the number line:

1 + 3

4

⇥

4

The y-coordinate is f(

1

) =

4

▶Strict local extrema:

{(1, 4)}