extreme x^4 — step-by-step solution

1) Find the strict local extrema of

x^{4}

on ℝ: the derivative must be zero and change sign

⇥1.1) Let's differentiate:

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

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

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

4 x^{3}

⇥

4 x^{3}

⇥The first derivative is

4 x^{3}

⇥1.2) Let's solve in

ℝ

4 x^{3} = 0

⇥We simplify the expression by

4

⇥⇥Let's solve in

ℝ

x^{3} = 0

⇥⇥

x^{3} = 0

means that:

x = 0

⇥⇥Solution found:

x = 0

⇥1.3) Let's perform the differentiation:

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

⇥ⓘ 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})

■The derivative of x^n is n*x^(n-1):

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

3 x^{2}

⇥

4 \cdot 3 x^{2}

⇥ⅰ

4 \cdot 3

12

⇥

12 x^{2}

⇥The second derivative is

12 x^{2}

2) At

x = 0

, calculate

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

The first nonzero derivative at this point has order 4 and value

24

The first nonzero derivative has even order and is positive: this is a strict local minimum

⇥

0^{4}

⇥

0

The y-coordinate is f(

0

) =

0

The polynomial tends to +∞ at both infinities: this unique local extremum is also global

▶Strict local extrema:

{(0, 0)}