Differentiable f:(−1,1)→ℝ satisfies: I:if f(0)≥0 and f′(x)>0 at every nonnegative zero, then f(x)>0 for x>0. II:if f(0)≤0 and f′(x)>0 at every nonpositive zero, then f(x)<0 for x<0. Which statements are correct?
For I, if f(0)=0 the positive derivative makes f positive immediately to the right; if f(0)>0 continuity does so. A first later zero after a positive interval would have derivative ≤0 from the left, contradicting the premise. Thus I holds. Apply the same argument to g(t)=−f(−t), whose derivative at zeros is positive, to obtain II. **C**.
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