Question

f(x)=x^4-8x^3+22x^2-24x+21 find the interval on which function is (a) increasing (b) decreasing

Answer 1

refer to the attachment

Answer 2

(a) (1, 2) U (3, ∞) (b) (-∞ , 1) U (2, 3)

it is given that,

f(x) = x⁴ - 8x³ + 22x² - 24x + 21

differentiating with respect to x,

f'(x) = 4x³ - 24x² + 44x - 24

= 4(x³ - 6x² + 11x - 6)

= 4(x³ - x² - 5x² + 5x + 6x - 6)

= 4(x - 1)(x² - 5x + 6)

= 4(x - 1)(x - 2)(x - 3)

(a) increasing,

f'(x) > 0

(x - 1)(x - 2)(x - 3) > 0

put x = 1, 2, 3 in number line and apply rule of inequality. we get, x ∈ (1, 2) U (3, ∞)

(b) decreasing

f'(x) < 0

(x - 1)(x - 2)(x - 3) < 0

x ∈ (-∞, 1) U (2, 3)

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