inflection x^5 — step-by-step solution

1) For

x^{5}

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

⇥1.1) Let's differentiate:

\frac{d}{d x} (x^{5})

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

\frac{d}{d x} (x^{5})

5 x^{4}

⇥

5 x^{4}

The first derivative is

5 x^{4}

⇥1.2) Let's perform the differentiation:

\frac{d}{d x} (5 x^{4})

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

\frac{d}{d x} (5 x^{4})

5 \cdot \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}

⇥

5 \cdot 4 x^{3}

⇥ⅰ

5 \cdot 4

20

⇥

20 x^{3}

The second derivative is

20 x^{3}

⇥1.3) Let's solve in

ℝ

20 x^{3} = 0

⇥We simplify the expression by

20

⇥⇥Let's solve in

ℝ

x^{3} = 0

⇥⇥

x^{3} = 0

is equivalent to:

x = 0

⇥⇥Solution obtained:

x = 0

⇥1.4) Let's perform the differentiation:

\frac{d}{d x} (20 x^{3})

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

\frac{d}{d x} (20 x^{3})

20 \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}

⇥

20 \cdot 3 x^{2}

⇥ⅰ

20 \cdot 3

60

⇥

60 x^{2}

⇥1.5) Let's differentiate:

\frac{d}{d x} (60 x^{2})

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

\frac{d}{d x} (60 x^{2})

60 \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

⇥

60 \cdot 2 x

⇥ⅰ

60 \cdot 2

120

⇥

120 x

⇥1.6) Let's perform the differentiation:

\frac{d}{d x} (120 x)

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

\frac{d}{d x} (120 x)

120

⇥

120

x = 0

, the zero of the second derivative has order 3: 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^{5}

⇥

0

The y-coordinate is f(

0

) =

0

▶Inflection points:

{(0, 0)}

🖼/draw?f1=x%5E5&v=x&b=-2.0&e=2.0&min=-38.4&max=38.4&points=0%2C0