inflection f(x)=x^4-54x^2+1 — step-by-step solution

1) For

x^{4} - 54 x^{2} + 1

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

⇥1.1) Let's differentiate:

\frac{d}{d x} (x^{4} - 54 x^{2} + 1)

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

\frac{d}{d x} (x^{4} - 54 x^{2} + 1)

\frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 54 x^{2}) + 0

⇥

\frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 54 x^{2}) + 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^{2})

2 x

■Adding 0 changes nothing:

\frac{d}{d x} (- 54 x^{2}) + 0

\frac{d}{d x} (- 54 x^{2})

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

\frac{d}{d x} (- 54 x^{2})

-54 \cdot \frac{d}{d x} (x^{2})

⇥

4 x^{3} - 54 \cdot 2 x

⇥ⓘ Multiplications are done before additions and subtractions:

54 \cdot 2

(50 + 4) \cdot 2

50 \cdot 2 + 4 \cdot 2

100 + 8

108

⇥

4 x^{3} - 108 x

The first derivative is

4 x^{3} - 108 x

⇥1.2) Let's differentiate:

\frac{d}{d x} (4 x^{3} - 108 x)

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

\frac{d}{d x} (4 x^{3} - 108 x)

\frac{d}{d x} (4 x^{3}) + \frac{d}{d x} (- 108 x)

⇥

\frac{d}{d x} (4 x^{3}) + \frac{d}{d x} (- 108 x)

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

\frac{d}{d x} (4 x^{3})

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

\frac{d}{d x} (- 108 x)

-108

■The derivative of x^n is n*x^(n-1):

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

3 x^{2}

⇥

4 \cdot 3 x^{2} - 108

⇥ⓘ Multiplications are done before additions and subtractions:

4 \cdot 3

12

⇥

12 x^{2} - 108

⇥We factor out

12

, the greatest common divisor of the coefficients:

12 x^{2} - 108

12 \cdot (x^{2} - 9)

⇥Factor identified:

x^{2} - 9

The second derivative is

12 \cdot (x^{2} - 9)

⇥1.3) Let's solve in

ℝ

12 \cdot (x^{2} - 9) = 0

⇥We simplify the expression by

12

⇥⇥Let's solve in

ℝ

x^{2} - 9 = 0

⇥⇥The two numbers whose square is

9

are

-3

and

3

⇥⇥Solutions obtained:

x = {-3, 3}

⇥1.4) Let's differentiate:

\frac{d}{d x} (12 \cdot (x^{2} - 9))

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

\frac{d}{d x} (12 \cdot (x^{2} - 9))

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

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

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

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

⇥

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

⇥ⓘ Adding 0 changes nothing:

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

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

■The derivative of x^n is n*x^(n-1):

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

2 x

⇥

12 \cdot 2 x

⇥ⅰ

12 \cdot 2

(10 + 2) \cdot 2

10 \cdot 2 + 2 \cdot 2

20 + 4

24

⇥

24 x

x = -3

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

Near

-3

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

⇥

{(-3)}^{4} - 54 \cdot {(-3)}^{2} + 1

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

{(-3)}^{4}

(-3) \cdot (-3) \cdot (-3) \cdot (-3)

81

{(-3)}^{2}

(-3) \cdot (-3)

9

⇥

81 - 54 \cdot 9 + 1

⇥ⓘ Multiplications are done before additions and subtractions:

54 \cdot 9

(50 + 4) \cdot 9

50 \cdot 9 + 4 \cdot 9

450 + 36

486

⇥

81 - 486 + 1

⇥ⓘ Start from

81

and move back

486

on the number line, going below zero:

81 - 486

-405

⇥

- 405 + 1

⇥ⓘ Start from

-405

and move forward

1

on the number line, getting closer to zero:

- 405 + 1

-404

⇥

-404

The y-coordinate is f(

-3

) =

-404

⇥1.5) Let's perform the differentiation:

\frac{d}{d x} (12 \cdot (x^{2} - 9))

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

\frac{d}{d x} (12 \cdot (x^{2} - 9))

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

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

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

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

⇥

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

⇥ⓘ Adding 0 changes nothing:

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

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

■The derivative of x^n is n*x^(n-1):

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

2 x

⇥

12 \cdot 2 x

⇥ⅰ

12 \cdot 2

(10 + 2) \cdot 2

10 \cdot 2 + 2 \cdot 2

20 + 4

24

⇥

24 x

x = 3

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

Near

3

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

⇥

3^{4} - 54 \cdot 3^{2} + 1

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

3^{4}

3 \cdot 3 \cdot 3 \cdot 3

81

3^{2}

3 \cdot 3

9

⇥

81 - 54 \cdot 9 + 1

⇥ⓘ Multiplications are done before additions and subtractions:

54 \cdot 9

(50 + 4) \cdot 9

50 \cdot 9 + 4 \cdot 9

450 + 36

486

⇥

81 - 486 + 1

⇥ⓘ Start from

81

and move back

486

on the number line, going below zero:

81 - 486

-405

⇥

- 405 + 1

⇥ⓘ Start from

-405

and move forward

1

on the number line, getting closer to zero:

- 405 + 1

-404

⇥

-404

The y-coordinate is f(

3

) =

-404

▶Inflection points:

{(-3, -404), (3, -404)}

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