extreme f(x)=(x^3)/3-2x^2+4x+3 — step-by-step solution

1) Find the strict local extrema of

\frac{x^{3}}{3}

- 2 x^{2}

+ 4 x

+ 3

on ℝ: the derivative must be zero and change sign

⇥1.1) Let's perform the differentiation:

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

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

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

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

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

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

+ 0

⇥

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

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

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

+ 0

⇥ⓘ A constant factor stays in front of the derivative, (k*u)' = k*u'. The derivative of x^n is n*x^(n-1):

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

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

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

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

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

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

4

■The derivative of x^n is n*x^(n-1):

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

2 x

■Adding 0 changes nothing:

\frac{d}{d x} (4 x) + 0

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

⇥

\frac{1}{3} \cdot 3 x^{2} - 2 \cdot 2 x + 4

⇥ⓘ Multiplications are done before additions and subtractions:

\frac{1}{3} \cdot 3

1

2 \cdot 2

4

■Multiplying by 1 changes nothing:

1 \cdot x^{2}

x^{2}

⇥

x^{2} - 4 x + 4

⇥The first derivative is

x^{2} - 4 x + 4

⇥1.2) Let's solve in

ℝ

x^{2} - 4 x + 4 = 0

⇥You can rewrite the expression as:

{(x - 2)}^{2}

⇥

{(x - 2)}^{2} = 0

is equivalent to:

x - 2 = 0

⇥ⓘ We add

2

to both sides

⇥Solution found:

x = 2

⇥1.3) Let's perform the differentiation:

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

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

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

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

⇥

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

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

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

2 x

■Adding 0 changes nothing:

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

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

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

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

-4

⇥

2 x - 4

⇥The second derivative is

2 x - 4

2) At

x = 2

, calculate

f^{''} (2) = 2 \cdot 2 - 4 = 0

x = 2

, the derivative has a zero of even order: it does not change sign, so this is not an extremum

This odd-degree polynomial is unbounded above and below on ℝ: it has no global extremum

▶Strict local extrema:

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