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Question:

The function f defined by f(x) = x³ - x² + 5x + 7, is decreasing in R, decreasing in (0, ∞) and increasing in (-∞, 0), increasing in (0, ∞) and decreasing in (-∞, 0), increasing in R

decreasing in R

increasing in (0, ∞) and decreasing in (-∞, 0)

decreasing in (0, ∞) and increasing in (-∞, 0)

increasing in R

Solution:

f(x) = x³ - x² + 5x + 7
f'(x) = 3x² - 2x + 5
The discriminant of the above quadratic equation is Δ = 36 - 4(3)(5) = 36 - 60 = -24 < 0
∴ f'(x) > 0 ∀x ∈ R⁺,
Also f'(x) > 0 ∀x ∈ R
∴ f is increasing in R.