extrema x^4-2x^2 — step-by-step solution

1) Find the strict local extrema of

x^{4} - 2 x^{2}

on ℝ: the derivative must be zero and change sign

⇥1.1) Let's perform the differentiation:

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

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

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

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

⇥

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

⇥ⓘ 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

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

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

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

⇥

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

⇥ⓘ Multiplications are done before additions and subtractions:

2 \cdot 2

4

⇥

4 x^{3} - 4 x

⇥The first derivative is

4 x^{3} - 4 x

⇥1.2) Let's solve in

ℝ

4 x^{3} - 4 x = 0

⇥You can write the expression:

4 x^{3} - 4 x

in factored form:

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

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

⇥⇥Let's solve in

ℝ

x = 0

⇥⇥Solution found:

x = 0

⇥⇥Let's solve in

ℝ

4 x^{2} - 4 = 0

⇥⇥ⓘ Both sides are divided by

4

⇥⇥

x^{2} = 1

⇥⇥The two numbers whose square is

1

are

-1

and

1

⇥⇥Solutions obtained:

x = {-1, 1}

⇥1.3) Let's perform the differentiation:

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

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

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

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

⇥

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

-4

■The derivative of x^n is n*x^(n-1):

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

3 x^{2}

⇥

4 \cdot 3 x^{2} - 4

⇥ⓘ Multiplications are done before additions and subtractions:

4 \cdot 3

12

⇥

12 x^{2} - 4

⇥The second derivative is

12 x^{2} - 4

2) At

x = -1

, calculate

f^{''} (-1) = 12 \cdot {(-1)}^{2} - 4 = 8

x = -1

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

⇥

{(-1)}^{4} - 2 \cdot {(-1)}^{2}

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

{(-1)}^{4}

(-1) \cdot (-1) \cdot (-1) \cdot (-1)

1

{(-1)}^{2}

(-1) \cdot (-1)

1

■Multiplying by 1 changes nothing:

2 \cdot 1

2

⇥

1 - 2

⇥ⓘ Start from

1

and move back

2

on the number line, going below zero:

1 - 2

-1

⇥

-1

The y-coordinate is f(

-1

) =

-1

3) At

x = 0

, calculate

f^{''} (0) = 12 \cdot 0^{2} - 4 = -4

x = 0

, the second derivative is negative: this is a strict local maximum

⇥

0^{4} - 2 \cdot 0^{2}

⇥ⓘ A product with a zero factor is zero:

-2 \cdot 0

0

■Adding 0 changes nothing:

0 + 0

0

⇥

0

The y-coordinate is f(

0

) =

0

4) At

x = 1

, calculate

f^{''} (1) = 12 \cdot 1^{2} - 4 = 8

x = 1

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

⇥

1^{4} - 2 \cdot 1^{2}

⇥

1 - 2

⇥ⓘ Start from

1

and move back

2

on the number line, going below zero:

1 - 2

-1

⇥

-1

The y-coordinate is f(

1

) =

-1

▶Strict local extrema:

{(-1, -1), (0, 0), (1, -1)}