inflection-x^4+12x^3-12x+13 — step-by-step solution

1) For

- x^{4}

+ 12 x^{3}

- 12 x

+ 13

on ℝ, find where the second derivative is zero and changes sign

⇥1.1) Let's perform the differentiation:

\frac{d}{d x} (- x^{4} + 12 x^{3} - 12 x + 13)

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

\frac{d}{d x} (- x^{4} + 12 x^{3} - 12 x + 13)

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

+ \frac{d}{d x} (12 x^{3})

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

+ 0

⇥

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

+ \frac{d}{d x} (12 x^{3})

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

+ 0

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

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

- 4 x^{3}

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

3 x^{2}

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

\frac{d}{d x} (12 x^{3})

12 \cdot \frac{d}{d x} (x^{3})

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

-12

■Adding 0 changes nothing:

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

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

⇥

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

⇥ⓘ Multiplications are done before additions and subtractions:

12 \cdot 3

(10 + 2) \cdot 3

10 \cdot 3 + 2 \cdot 3

30 + 6

36

⇥

- 4 x^{3} + 36 x^{2} - 12

⇥We factor out

4

, the greatest common divisor of the coefficients:

- 4 x^{3} + 36 x^{2} - 12

4 \cdot (- x^{3} + 9 x^{2} - 3)

⇥Factor found:

- x^{3} + 9 x^{2} - 3

The first derivative is

4 \cdot (- x^{3} + 9 x^{2} - 3)

⇥1.2) Let's differentiate:

\frac{d}{d x} (4 \cdot (- x^{3} + 9 x^{2} - 3))

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

\frac{d}{d x} (4 \cdot (- x^{3} + 9 x^{2} - 3))

4 \cdot \frac{d}{d x} (- x^{3} + 9 x^{2} - 3)

■The derivative of a sum is the sum of the derivatives:

\frac{d}{d x} (- x^{3} + 9 x^{2} - 3)

\frac{d}{d x} (- x^{3}) + \frac{d}{d x} (9 x^{2}) + 0

⇥

4 \cdot (\frac{d}{d x} (- x^{3}) + \frac{d}{d x} (9 x^{2}) + 0)

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

\frac{d}{d x} (- x^{3})

- 3 x^{2}

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

2 x

■Adding 0 changes nothing:

\frac{d}{d x} (9 x^{2}) + 0

\frac{d}{d x} (9 x^{2})

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

\frac{d}{d x} (9 x^{2})

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

⇥

4 \cdot (- 3 x^{2} + 9 \cdot 2 x)

⇥ⓘ Multiplications are done before additions and subtractions:

9 \cdot 2

18

⇥

4 \cdot (- 3 x^{2} + 18 x)

⇥

4 \cdot (18 x - 3 x^{2})

The second derivative is

4 \cdot (18 x - 3 x^{2})

⇥1.3) Let's solve in

ℝ

4 \cdot (18 x - 3 x^{2}) = 0

⇥We simplify the expression by

4

⇥⇥Let's solve in

ℝ

18 x - 3 x^{2} = 0

⇥⇥This expression can be factorized into:

x \cdot (18 - 3 x)

⇥⇥A product is zero if and only if one of its factors is zero: we solve each factor equal to 0

⇥⇥⇥1.3.1) Let's solve in

ℝ

x = 0

⇥⇥⇥Solution found:

x = 0

⇥⇥⇥1.3.2) Let's solve in

ℝ

18 - 3 x = 0

⇥⇥⇥

3 x = 18

⇥⇥⇥ⓘ Both sides are divided by

3

⇥⇥⇥

x = \frac{18}{3}

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

3

, their greatest common divisor:

\frac{18}{3}

6

⇥⇥⇥Solution found:

x = 6

⇥1.4) Let's differentiate:

\frac{d}{d x} (4 \cdot (18 x - 3 x^{2}))

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

\frac{d}{d x} (4 \cdot (18 x - 3 x^{2}))

4 \cdot \frac{d}{d x} (18 x - 3 x^{2})

■The derivative of a sum is the sum of the derivatives:

\frac{d}{d x} (18 x - 3 x^{2})

\frac{d}{d x} (18 x) + \frac{d}{d x} (- 3 x^{2})

⇥

4 \cdot (\frac{d}{d x} (18 x) + \frac{d}{d x} (- 3 x^{2}))

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

\frac{d}{d x} (18 x)

18

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

-3 \cdot \frac{d}{d x} (x^{2})

■The derivative of x^n is n*x^(n-1):

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

2 x

⇥

4 \cdot (18 - 3 \cdot 2 x)

⇥ⓘ Multiplications are done before additions and subtractions:

3 \cdot 2

6

⇥

4 \cdot (18 - 6 x)

⇥Let's factorize in

ℝ

(18 - 6 x)

⇥We factor out

6

, the greatest common divisor of the coefficients:

18 - 6 x

6 \cdot (3 - x)

⇥Factor found:

3 - x

⇥

4 \cdot 6 \cdot (3 - x)

⇥ⅰ

4 \cdot 6

24

⇥

24 \cdot (3 - x)

x = 0

, the zero of the second derivative has order 1: this order is odd, so the concavity changes

Near

0

, the second derivative is negative on the left and positive on the right: the curve changes from concave down to concave up

⇥

- 0^{4}

+ 12 \cdot 0^{3}

- 12 \cdot 0

+ 13

⇥ⓘ A product with a zero factor is zero:

12 \cdot 0

0

■Adding 0 changes nothing:

0 + 13

13

⇥

0 + 0 + 13

⇥ⓘ Adding 0 changes nothing:

0 + 0 + 13

13

⇥

13

The y-coordinate is f(

0

) =

13

⇥1.5) Let's perform the differentiation:

\frac{d}{d x} (4 \cdot (18 x - 3 x^{2}))

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

\frac{d}{d x} (4 \cdot (18 x - 3 x^{2}))

4 \cdot \frac{d}{d x} (18 x - 3 x^{2})

■The derivative of a sum is the sum of the derivatives:

\frac{d}{d x} (18 x - 3 x^{2})

\frac{d}{d x} (18 x) + \frac{d}{d x} (- 3 x^{2})

⇥

4 \cdot (\frac{d}{d x} (18 x) + \frac{d}{d x} (- 3 x^{2}))

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

\frac{d}{d x} (18 x)

18

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

-3 \cdot \frac{d}{d x} (x^{2})

■The derivative of x^n is n*x^(n-1):

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

2 x

⇥

4 \cdot (18 - 3 \cdot 2 x)

⇥ⓘ Multiplications are done before additions and subtractions:

3 \cdot 2

6

⇥

4 \cdot (18 - 6 x)

⇥Let's factorize in

ℝ

(18 - 6 x)

⇥We factor out

6

, the greatest common divisor of the coefficients:

18 - 6 x

6 \cdot (3 - x)

⇥Factor found:

3 - x

⇥

4 \cdot 6 \cdot (3 - x)

⇥ⅰ

4 \cdot 6

24

⇥

24 \cdot (3 - x)

x = 6

, the zero of the second derivative has order 1: this order is odd, so the concavity changes

Near

6

, the second derivative is positive on the left and negative on the right: the curve changes from concave up to concave down

⇥

- 6^{4}

+ 12 \cdot 6^{3}

- 12 \cdot 6

+ 13

⇥ⓘ Without parentheses, the minus sign is not part of the power■An integer power is a repeated product:

- 6^{4}

- (6 \cdot 6 \cdot 6 \cdot 6)

-1296

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

6^{3}

6 \cdot 6 \cdot 6

216

■Multiplications are done before additions and subtractions:

12 \cdot 6

(10 + 2) \cdot 6

10 \cdot 6 + 2 \cdot 6

60 + 12

72

⇥

- 1296

+ 12 \cdot 216

- 72

+ 13

⇥ⓘ Multiplications are done before additions and subtractions:

12 \cdot 216

(10 + 2) \cdot 216

10 \cdot 216 + 2 \cdot 216

2160 + 432

2592

⇥

- 1296

+ 2592

- 72

+ 13

⇥ⓘ Start from

-1296

and move forward

2592

on the number line, going past zero:

- 1296 + 2592

1296

⇥

1296 - 72 + 13

⇥ⓘ Start from

1296

and move back

72

on the number line:

1296 - 72

1224

⇥

1224 + 13

⇥ⓘ Start from

1224

and move forward

13

on the number line:

1224 + 13

1237

⇥

1237

The y-coordinate is f(

6

) =

1237

▶Inflection points:

{(0, 13), (6, 1237)}

🖼/draw?f1=-x%5E4+%2B+12*x%5E3+-+12*x+%2B+13&v=x&b=-3.0&e=9.0&min=-600.8&max=2336.8&points=0%2C13%3B6%2C1237