inflection points of x^3+2x^2+x-7

1) For

x^{3}

+ 2 x^{2}

+ x

- 7

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

⇥1.1) Let's differentiate:

\frac{d}{d x} (x^{3} + 2 x^{2} + x - 7)

⇥

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

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

+ 1

+ 0

⇥ℹ

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

3 x^{2}

‖

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

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

‖

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

2 x

⇥

3 x^{2} + 2 \cdot 2 x + 1

⇥

3 x^{2} + 4 x + 1

The first derivative is

3 x^{2} + 4 x + 1

⇥1.2) Let's differentiate:

\frac{d}{d x} (3 x^{2} + 4 x + 1)

⇥

\frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (4 x) + 0

⇥ℹ

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

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

‖

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

2 x

‖

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

4

⇥

3 \cdot 2 x + 4

⇥

6 x + 4

The second derivative is

6 x + 4

⇥1.3) Let's solve in

ℝ

6 x + 4 = 0

⇥

6 x = -4

⇥

x = - \frac{4}{6}

⇥Solution found:

x = - \frac{2}{3}

⇥1.4) Let's differentiate:

\frac{d}{d x} (6 x + 4)

⇥

\frac{d}{d x} (6 x) + 0

⇥

6

x = - \frac{2}{3}

, the zero of the second derivative has order 1: this order is odd, so the concavity changes

Near

- \frac{2}{3}

, the second derivative is negative on the left and positive on the right: the curve changes from concave down to concave up

⇥

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

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

- \frac{2}{3}

- 7

⇥

{(- \frac{2}{3})}^{3} + 2 \cdot {(- \frac{2}{3})}^{2} + \frac{- 2 - 7 \cdot 3}{3}

⇥

{(- \frac{2}{3})}^{3} + 2 \cdot {(- \frac{2}{3})}^{2} + \frac{- 2 - 21}{3}

⇥

{(- \frac{2}{3})}^{3} + 2 \cdot {(- \frac{2}{3})}^{2} - \frac{23}{3}

⇥ℹ

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

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

‖

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

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

⇥

\frac{{(-2)}^{3}}{3^{3}} + 2 \cdot \frac{{(-2)}^{2}}{3^{2}} - \frac{23}{3}

⇥ℹ

{(-2)}^{3}

-8

‖

3^{3}

27

‖

{(-2)}^{2}

4

‖

3^{2}

9

⇥

- \frac{8}{27} + 2 \cdot \frac{4}{9} - \frac{23}{3}

⇥ℹ

2 \cdot \frac{4}{9}

\frac{8}{9}

‖

- \frac{8}{27} - \frac{23}{3}

\frac{- 8 - 23 \cdot 9}{27}

⇥

\frac{- 8 - 23 \cdot 9}{27} + \frac{8}{9}

⇥

\frac{- 8 - 207}{27} + \frac{8}{9}

⇥

- \frac{215}{27} + \frac{8}{9}

⇥

\frac{- 215 + 8 \cdot 3}{27}

⇥

\frac{- 215 + 24}{27}

⇥

- \frac{191}{27}

The y-coordinate is f(

- \frac{2}{3}

) =

- \frac{191}{27}

▶Inflection points:

{(- \frac{2}{3}, - \frac{191}{27})}

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