inflection f(x)=3-5x^4 — step-by-step solution

1) For

3 - 5 x^{4}

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

⇥1.1) Let's perform the differentiation:

\frac{d}{d x} (3 - 5 x^{4})

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

\frac{d}{d x} (3 - 5 x^{4})

0 + \frac{d}{d x} (- 5 x^{4})

⇥

0 + \frac{d}{d x} (- 5 x^{4})

⇥ⓘ Adding 0 changes nothing:

0 + \frac{d}{d x} (- 5 x^{4})

\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 first derivative is

- 20 x^{3}

⇥1.2) Let's differentiate:

\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}

The second derivative is

- 60 x^{2}

⇥1.3) Let's solve in

ℝ

- 60 x^{2} = 0

⇥We simplify the expression by

-60

⇥⇥Let's solve in

ℝ

x^{2} = 0

⇥⇥

x^{2} = 0

means that:

x = 0

⇥⇥Solution obtained:

x = 0

⇥1.4) Let's perform the differentiation:

\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

⇥1.5) Let's differentiate:

\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 2: this order is even, so the concavity does not change: this is not an inflection point

▶Inflection points:

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🖼/draw?f1=3+-+5*x%5E4&v=x&b=-2.0&e=2.0&min=-85.0&max=11.0