extreme f(x)=5x^2+6x-7 — step-by-step solution

1) Find the strict local extrema of

5 x^{2} + 6 x - 7

on ℝ: the derivative must be zero and change sign

⇥1.1) Let's differentiate:

\frac{d}{d x} (5 x^{2} + 6 x - 7)

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

\frac{d}{d x} (5 x^{2} + 6 x - 7)

\frac{d}{d x} (5 x^{2}) + \frac{d}{d x} (6 x) + 0

⇥

\frac{d}{d x} (5 x^{2}) + \frac{d}{d x} (6 x) + 0

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

\frac{d}{d x} (5 x^{2})

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

\frac{d}{d x} (6 x)

6

■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} (6 x) + 0

\frac{d}{d x} (6 x)

⇥

5 \cdot 2 x + 6

⇥ⓘ Multiplications are done before additions and subtractions:

5 \cdot 2

10

⇥

10 x + 6

⇥The first derivative is

10 x + 6

⇥1.2) Let's solve in

ℝ

10 x + 6 = 0

⇥ⓘ We subtract

6

from both sides

⇥

10 x = -6

⇥ⓘ Both sides are divided by

10

⇥

x = - \frac{6}{10}

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

2

, their greatest common divisor:

- \frac{6}{10}

- \frac{3 \cdot 2}{5 \cdot 2}

- \frac{3}{5}

⇥Solution found:

x = - \frac{3}{5}

⇥1.3) Let's differentiate:

\frac{d}{d x} (10 x + 6)

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

\frac{d}{d x} (10 x + 6)

\frac{d}{d x} (10 x) + 0

⇥

\frac{d}{d x} (10 x) + 0

⇥ⓘ Adding 0 changes nothing:

\frac{d}{d x} (10 x) + 0

\frac{d}{d x} (10 x)

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

\frac{d}{d x} (10 x)

10

⇥

10

⇥The second derivative is

10

2) At

x = - \frac{3}{5}

, calculate

f^{''} (- \frac{3}{5}) = 10

x = - \frac{3}{5}

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

⇥

5 \cdot {(- \frac{3}{5})}^{2} + 6 \cdot (- \frac{3}{5}) - 7

⇥ⓘ The minus sign is placed in front of the product:

6 \cdot (- \frac{3}{5})

- 6 \cdot \frac{3}{5}

⇥

5 \cdot {(- \frac{3}{5})}^{2} - 6 \cdot \frac{3}{5} - 7

⇥ⓘ Multiplying a fraction by a number multiplies its numerator:

6 \cdot \frac{3}{5}

\frac{6 \cdot 3}{5}

\frac{18}{5}

⇥

5 \cdot {(- \frac{3}{5})}^{2} - \frac{18}{5} - 7

⇥ⓘ To add fractions, put them over the same denominator, then add the numerators:

- \frac{18}{5} - 7

\frac{- 18 - 7 \cdot 5}{5}

⇥

5 \cdot {(- \frac{3}{5})}^{2} + \frac{- 18 - 7 \cdot 5}{5}

⇥ⓘ Multiplications are done before additions and subtractions:

7 \cdot 5

35

⇥

5 \cdot {(- \frac{3}{5})}^{2} + \frac{- 18 - 35}{5}

⇥ⓘ The numerator and the denominator are computed before the division:

- 18 - 35

-53

⇥

5 \cdot {(- \frac{3}{5})}^{2} - \frac{53}{5}

⇥ⓘ The power of a quotient is the quotient of the powers, (a/b)^n = a^n/b^n:

{(- \frac{3}{5})}^{2}

\frac{{(-3)}^{2}}{5^{2}}

⇥

5 \cdot \frac{{(-3)}^{2}}{5^{2}} - \frac{53}{5}

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

{(-3)}^{2}

(-3) \cdot (-3)

9

5^{2}

5 \cdot 5

25

⇥

5 \cdot \frac{9}{25} - \frac{53}{5}

⇥ⓘ Multiplying a fraction by a number multiplies its numerator:

5 \cdot \frac{9}{25}

\frac{5 \cdot 9}{25}

\frac{45}{25}

⇥

\frac{45}{25} - \frac{53}{5}

⇥ⓘ To add fractions, put them over the same denominator, then add the numerators. We choose

25

, the least common multiple of

25

and

5

. Multiplying the numerator and the denominator by the same number does not change the fraction:

\frac{45}{25} - \frac{53}{5}

\frac{45}{25} + \frac{-53 \cdot 5}{5 \cdot 5}

\frac{45 - 53 \cdot 5}{25}

⇥

\frac{45 - 53 \cdot 5}{25}

⇥ⓘ Multiplications are done before additions and subtractions:

53 \cdot 5

(50 + 3) \cdot 5

50 \cdot 5 + 3 \cdot 5

250 + 15

265

⇥

\frac{45 - 265}{25}

⇥ⓘ The numerator and the denominator are computed before the division:

45 - 265

-220

⇥

- \frac{220}{25}

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

5

, their greatest common divisor:

- \frac{220}{25}

- \frac{44 \cdot 5}{5 \cdot 5}

- \frac{44}{5}

⇥

- \frac{44}{5}

The y-coordinate is f(

- \frac{3}{5}

) =

- \frac{44}{5}

▶Strict local extrema:

{(- \frac{3}{5}, - \frac{44}{5})}