Відповіді АЛГЕБРА: 22.13 (1-2)

22.13

1) f(x) = x² • (x – 4)²

f‘(x) = 2x • (x – 4)² + x² • 2(x – 4) = 2x • (x – 4)(x – 4 + x) = 2x(x –

– 4)(2x – 4);

2x • (x – 4)(2x – 4) = 0;

x = 0; x = 4; x = 2;

Xmin = 0; Xmin = 4; fmin = 0;

Xmax = 2; fmax = 4 • (2 – 4)² = 4 • 4 = 16.

2) g(x) = – 1x2- 4x+5 g(x) = –(x² – 4x + 5)^–1

D(y): x² – 4x + 5 ≠ 0;

D = 16 – 4 • 5 < 0;

∅ x є R;

g’(x) = (x² – 4x + 5)^–2 • (2x – 4);

g’(x) = 0;

2x – 4 = 0;

x = 2;

Xmin = 2;

gmin = –14-8+5 = –1.

3) t(x) = x2- 1x2- 4

D(y): x ≠ ± 2;

t’(x) = 2xx2-4- 2x(x2- 1)(x2- 4)2 = 2x3- 8x-2x3+ 2x(x2- 4)2 = -6x(x2- 4)2

t’(x) = 0;

x = 0;

Xmax = 0;

ymax = 0-10-4 = 14

4) Ψ(x) = x+2x2- 9

D(y): x ≠ ± 3;

Ψ(x) = x2- 9-2x (x+2)(x2- 9)2 = x2- 9-2x-4x(x2- 9)2 = -x2- 4x-9(x2- 9)2

Ψ’(x) = 0;

–x² – 4x – 9 = 0;

x² + 4x + 9 = 0;

D = 16 – 4 • 9 < 0;

∅ Екстремумів нема.