critical f(x)=4x^3+7x^2-20x+9 — step-by-step solution

⚠Ⓘ 4x is read as the product 4*x: for a power, write 4^x‖7x is read as the product 7*x: for a power, write 7^x‖20x is read as the product 20*x: for a power, write 20^x

1) A polynomial is differentiable on ℝ: its critical points are the zeros of its derivative

⇥1.1) Let's differentiate:

\frac{d}{d x} (4 x^{3} + 7 x^{2} - 20 x + 9)

⇥

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

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

+ \frac{d}{d x} (- 20 x)

+ 0

⇥ℹ

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

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

‖

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

3 x^{2}

‖

\frac{d}{d x} (7 x^{2})

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

‖

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

2 x

‖

\frac{d}{d x} (- 20 x)

-20

⇥

4 \cdot 3 x^{2} + 7 \cdot 2 x - 20

⇥ℹ

4 \cdot 3

12

‖

7 \cdot 2

14

⇥

12 x^{2} + 14 x - 20

⇥1.2) Let's solve in

ℝ

12 x^{2} + 14 x - 20 = 0

⇥You can factorize this expression into:

2 \cdot (6 x^{2} + 7 x - 10) = 0

⇥1.3) Let's solve:

6 x^{2} + 7 x - 10 = 0

⇥Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

⇥

Δ = 7^{2} - 4 \cdot 6 \cdot (-10)

⇥ℹ

7^{2}

49

‖

4 \cdot 6

24

⇥

Δ = 49 + 24 \cdot 10

⇥

Δ = 49 + 240

⇥

Δ = 289

⇥As

Δ > 0

, the equation has two solutions:

⇥The first solution is:

\frac{- b - \sqrt{Δ}}{2 a}

⇥

x1 = \frac{- 7 - \sqrt{289}}{2 \cdot 6}

⇥ℹ

- \sqrt{289}

-17

‖

2 \cdot 6

12

⇥

x1 = \frac{- 7 - 17}{12}

⇥

x1 = - \frac{24}{12}

⇥We simplify the fraction:

\frac{24}{12}

by the common factor:

12

to get the new expression:

2

⇥

x1 = -2

⇥For the second solution, we obtain:

\frac{- b + \sqrt{Δ}}{2 a}

⇥

x2 = \frac{- 7 + \sqrt{289}}{2 \cdot 6}

⇥ℹ

\sqrt{289}

17

‖

2 \cdot 6

12

⇥

x2 = \frac{- 7 + 17}{12}

⇥

x2 = \frac{10}{12}

⇥

x2 = \frac{5}{6}

⇥Solution found:

x = -2

⇥Solution found:

x = \frac{5}{6}

▶Critical points:

x = {-2, \frac{5}{6}}