Question

Найдите наибольшее и наименьшее значения функции на заданном отрезке.
у = х^4 – 8х^3 +10х^2 + 1 на [-1; 7];

Answer 1

1)y = x⁴-8x³+10x²+1, [-1; 7]

y' = 4x³-24x²+20x

2)4x³-24x²+20x = 0

4x(x²-6x+5) = 0

4x(x²-5x-x+5) = 0

4x(x(x-5)-(x-5)) = 0

4x(x-5)(x-1) = 0

x = 0; 1; 5

3)f(-1) = 1+8+10+1 = 20

f(0) = 1

f(1) = 1-8+10+1 = 4

f(5) = 5⁴-8•5³+10•5²+1 = 5²(5²-8•5+10)+1 = 25(25-40+10)+1 = -125+1 = -124, min

f(7) = 7⁴-8•7³+10•7²+1 = 7²(7²-8•7+10)+1 = 49(49-56+10)+1 = 147, max