§ 41: 41.30

41.30 Розв’яжіть рівняння

x+7 + 1-x = x² + 6x + 13.

f(x) = x+7 + 1-x; g(x) = x² + 6x + 13; [–7; –1]

1) f’(x) = 12x+7 + -121-7 = 1-x2x+7 – x+721-x = 1-x- x+72x+7(1-x).

f’(x) = 0; 1-x = x+7; 1 – x = x + 7; 2x = –6; x = –3.

f(–3) = -3+7 + 1+3 = 2 + 2 = 4.

2) g’(x) = 2x + 6; g’(x) = 0; 2x = –6; x = –3.

g(–3) = (–3)² – 6 • 3 + 13 = 9 – 18 + 13 = 4.

3) f(–7) = -7+7 + 1+7 = 0 + 8 = 22; f(–1) = -1+7 + 1+1 = 6 + 2;

g(–7) = 49 – 42 + 13 = 20; g(–1) = 1 – 6 + 13 = 8.

Маємо: f(x) ≤ 4; g(x) ≥ 4. Розв‘язком рівняння є х = –3.