extreme x^3 — step-by-step solution

1) Find the strict local extrema of

x^{3}

on ℝ: the derivative must be zero and change sign

⇥1.1) Let's differentiate:

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

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

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

3 x^{2}

⇥

3 x^{2}

⇥The first derivative is

3 x^{2}

⇥1.2) Let's solve in

ℝ

3 x^{2} = 0

⇥We simplify the expression by

3

⇥⇥Let's solve in

ℝ

x^{2} = 0

⇥⇥

x^{2} = 0

is equivalent to:

x = 0

⇥⇥Solution found:

x = 0

⇥1.3) Let's perform the differentiation:

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

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

\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

⇥

3 \cdot 2 x

⇥ⅰ

3 \cdot 2

6

⇥

6 x

⇥The second derivative is

6 x

2) At

x = 0

, calculate

f^{''} (0) = 6 \cdot 0 = 0

x = 0

, the derivative has a zero of even order: it does not change sign, so this is not an extremum

This odd-degree polynomial is unbounded above and below on ℝ: it has no global extremum

▶Strict local extrema:

\emptyset