inflection f(x)=1/(x-2) — step-by-step solution

⇥1) Let's find the domain of

\frac{1}{x - 2}

⇥ For the expression

\frac{1}{x - 2}

to exist, you must check:

x - 2 \neq 0

⇥ⓘ We add

2

to both sides

⇥Solution found:

x \neq 2

An inflection point must lie on the curve: stay within the domain

x \neq 2

⇥To simplify differentiation, we rewrite

\frac{1}{x - 2}

{(x - 2)}^{-1}

⇥2) Let's differentiate:

\frac{d}{d x} ({(x - 2)}^{-1})

⇥Apply the differentiation rule:

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{n - 1}

with f =

x - 2

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

\frac{d}{d x} ({(x - 2)}^{-1})

-1 \cdot \frac{d}{d x} (x - 2) \cdot {(x - 2)}^{-2}

⇥

-1 \cdot \frac{d}{d x} (x - 2) \cdot {(x - 2)}^{-2}

⇥ⓘ Multiplying by -1 gives the opposite:

-1 \cdot \frac{d}{d x} (x - 2) \cdot {(x - 2)}^{-2}

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

⇥

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

⇥ⓘ The factor is moved to the numerator or the denominator by changing the sign of its exponent, for a nonzero base:

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

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

⇥

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

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

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

1 + 0

⇥

- \frac{1 + 0}{{(x - 2)}^{2}}

⇥ⓘ Adding 0 changes nothing:

1 + 0

1

⇥

- \frac{1}{{(x - 2)}^{2}}

⇥To simplify differentiation, we rewrite

- \frac{1}{{(x - 2)}^{2}}

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

⇥3) Let's differentiate:

\frac{d}{d x} (-1 \cdot {(x - 2)}^{-2})

⇥ⓘ Multiplying by -1 gives the opposite:

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

- {(x - 2)}^{-2}

⇥

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

⇥We differentiate using:

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{n - 1}

(where f =

x - 2

)

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

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

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

⇥

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

⇥ⓘ Extract the minus signs: an even number of minus signs cancels out, an odd number leaves one minus sign:

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

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

⇥

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

⇥ⓘ The factor is moved to the numerator or the denominator by changing the sign of its exponent, for a nonzero base:

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

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

⇥

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

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

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

1 + 0

⇥

\frac{2 \cdot (1 + 0)}{{(x - 2)}^{3}}

⇥ⓘ Adding 0 changes nothing:

1 + 0

1

⇥

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

The second derivative

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

is identically zero or never zero on each domain interval. The excluded pole is not an inflection point

▶Inflection points:

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