Convessità e ricerca di massimi e minimi da esame

Concavità, convessità, massimi e minimi

1) f(x) = x - √2 - x Concavità

2) f(x) = x (x - 1) (Max e min) f(x) = ln (x² - 25)

3) f(x) = x²e^x

4) f(x) = 1/x + ln x

1) f(x) = x - √2x Concavità _EMRE + Max & min

2) f(x) = x(x-1) (Max & min) f(x) = ln(x²-25)

3) f(x) = x²e^2x

4) f(x) = 1/x + ln x

Convessità e.m.e. + max & min

• F(x) = x - √2-x
• F(x) = x(x-1) (max e min)
• F(x) = x2^x
• F(x) = ¹⁄_x + ln x
• F(x) = (3x² + 1) ⁄ x²
• F(x) = x ln (x² - 25)
• F(x) = x¹e^ux⁄₂
• F(x) = √x²-(x²-x+1)⁄_x³-1
• F(x) = x ln (1+√x²)
• F(x) = arctan(1+x²)
• E^x-(x²-x)
• F(x) = e^x(x2+x)

Funzione 1

f(x) = x - √2-x

f'(x) = 1 + ¹⁄_2√2-x

f''(x) = ^.1⁄_4√2-x(2-x)

¹⁄_4√2-x(2-x) > 0

√2-x (2-x) > 0

x < 2

Min e max

f(x) = x(x-1)

f''(x) = 2x-1

f'(x) ≠ 0

2x-1 > ¹⁄₂

Convessità

f(x) = ln(x²-25)

f'(x) = ^2x⁄_(x²-25)

f''(x) = ^-2x-50⁄_(x²-25)²

(x²-50) (x²-25)² > 0

-2x²-50 ⁄ (x²-25)² > 0

2x-50 ⁄ (x²-25)² < 0

x = x+5, x≠-5

x ≤ 5

x ≤ x∈R

Funzione 3

f'(x) = x²e^x

f''(x) = 2 xe^x + x²e^x

f'''(x) = 2e^x + 2xe^x + 2xe^x + x²e^x

f'''(x) = e^x(x² + 4x + 2)

x² + 4x + 2 ≥ 0

^{-4 ± √8} /₂ ➜ -2 ± √2

x + (-2 + √2))(x - (-2 - √2)) ≥ 0

x + 2 - √2 > 0

x + 2 + √2 > 0

x > -2 + √2

x > -2 - √2

➜ [-2 + √2 , +∞)

x + 2 - √2 ≤ 0

x + 2 + √2 ≤ 0

x < -2 + √2

x < -2 - √2

➜ (-∞, -2 - √2]

-∞, -2 - √2] U [-2 + √2, +∞

Funzione 4

f(x) = ¹/_x + ln x

f'(x) = -¹/_x² + ¹/_x

f''(x) = ²/_x³ - ¹/_x²

^{2 - x}/_x³ ➙

x ≤ 2

x > 0

➜(0,2]

2 - x < 0

x³ < 0

x > 2

x < 0

Impossible

(0,2]

Funzione 5

f(x) = (3x² + 4x + 1) e^x2

f'(x) = e^x2(3x² + 6x + 1)

f''(x) = e^x2(3x² + 12x + 7)

3x² + 12x + 7 ≥ 0

^{-12 + 2√15}/₆ ➜^-7/₃

^{-12 - 2√15}/₆

(x - x₁) (x - x₂) ≥ 0

(x - ^{-12 + 2√15}/₆) ≥ 0

(x - ^{-12 - 2√15}/₆) ≥ 0

x ≥ ^{-12 + 2√15}/₆

x ≥ ^{-12 - 2√15}/₆

➜ x > x ≥ ^{-12 - 2√15}/₆

(-∞ , -^{12 - 2√15}/₆] U [-^{12 + 2√15}/₆, +∞)

f(x) = x ln x - x²

f'(x) = ln x + 1 - 2x

f''(x) = ^{1 - 2x}/_x

^{1 - 2x}/_x > 0

{ 1 - 2x > 0 { x ≤ ¹/₂

x > 0 x > 0

(0, ¹/₂]

^{1 - 2x}/_x < 0

{ 1 - 2x < 0 { x ≥ ¹/₂

x < 0 x < 0

IMP

f(x) = ^{x2 ln x}/₂

f'(x) = ¹/₂ x² ln x

f''(x) = ¹/₂ (2x ln x + x) = x ln x + ^x/₂

f''(x) = ln x + ³/₂

ln x + ³/₂ > 0

ln x > -³/₂

x > e^-3/2

Pt. 1

f(x) = x(x² − x−1)

f'(x) = 3x² − 2x − 1 > 0

(x−1)(x−(1/3)) > 0

x−1 > 0

x > 1→ 1 , +∞

x−(1/3) > 0

x > − 1/3→ −∞, −1/3

−1/3 | 1

+−+

Pt. 2

f(x) = √(2^x − 1)→ f'(x) = 2^x / √(2^x−1)→ f''(x) = (2*4^x−2*2^2x)/√(2^x−1)

f''(x) = 2^x*(2^x−2) / (2^x−1)(2^x−1) > 0→ f''(x) < 0

2^x ≥ 2

2x ≥ ln 2

x ≥ ln 2/2→ [1/2 ln 2, +∞)

→ (−∞,0)

x ≤ ln 2/2

x < 0→ (−∞,0)

Inters. (−∞, 0) ∪ [1/2 ln 2, +∞)∩

f(x) = x ln x (1 + x²)

f '(x) = ln(1 + x²) + 2 x² / 1 + x²

f ''(x) = 6 x * 1, 2 x 3 / (1 + x²)²

(6x + 2x + 3) / (1 + x²)² > 0

↓2x(3x + 2) / (1 + x²)² > 0→ 3x x² > 0

1 + x² > 0 → sempre

{ x^t > -3 → sempre

10

f(x) = arcotan (1 + x²)

f '(x) = -2x / x⁴ + 2 x² + 2

f ''(x) = -6x⁴ - 6x² + 4 / (x⁴ + 2x² + 2)² > 0→ -6x⁴ - 6x² + 4 / (x⁴ + 2x² + 2)² → sempre > 0

-6t² - 4t + 4 > 0

↓(x + (1 + √7) / 3)(x + (1 - √7) / 3) > 0

x > (1 + √7) / 3

x > - (1 - √7) / 3 → [- (1 - √7) / 3, + ∞)

x < - (1 + √7) / 3

x < - (1 + √7) / 3 → (-∞, - (1 + √7) / 3]

Inters. [- (1 - √7) / 3, - (1 + √7) / 3]