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Question:

Let a ∈ R and let f: R → R be given by f(x) = x⁵ - x + a. Then f(x) has only one real root if a > 4; f(x) has three real roots if a < -4; f(x) has three real roots if -4 < a < 4.

f(x) has three real roots if -4 < a < 4

f(x) has three real roots if a < -4

f(x) has three real roots if a > 4

f(x) has only one real root if a > 4

Solution:

Let f(x) = x⁵ - x + a
f'(x) = 5x⁴ - 1
f'(x) = 0 ⇒ x = ±1
f''(x) = 20x³
Hence, x = 1 will be a point of local minimum and x = -1 will be a point of local maximum.
f(1) = -4 + a, f(-1) = 4 + a.