f(x) = ∫_(x+1\()^{0}\) (\(t^{3}\) - t)dt + ∫_0^(x + 1) (\(t^{3}\) - t)dt
= -∫_0^(x + 1) (\(t^{3}\) - t)dt + ∫_\(0^{x}\) (\(t^{3}\) - t)dt
より
f'(x) = -((x + 1\()^{3}\) - (x + 1)) + \(x^{3}\) - x
= -\(x^{3}\) - 3\(x^{2}\) - 3x - 1 + x + 1 + \(x^{3}\) - x
= -3\(x^{2}\) -3x = -3x(x + 1)
より
x = 0 で極小
f(0) = ∫_\(1^{0}\) (\(t^{3}\) - t)dt = [\(t^{4}\)/4 - \(t^{2}\)/2]_\(1^{0}\)
= -(\(\frac{1}{4}\) - \(\frac{1}{2}\)) = \(\frac{1}{4}\).
x = -1 で極大
f(-1) = ∫_0^(-1)(\(t^{3}\) - t)dt = [\(t^{4}\)/4 - \(t^{2}\)/2]_0^(-1)
= \(\frac{1}{4}\) - \(\frac{1}{2}\) = -\(\frac{1}{4}\).

g(t):=\(t^{3}\)-t と置けば、f'(x)=g(x)-g(x+1)=-3x(x+1).

f(x)をxで微分して、増減表を作れば良い。