Let \(f:\mathbb{R}\to\mathbb{R}\) be a function and let \(a\) be a real number. Let \(S\) be the statement “If \(f\) has a local minimum at \(a\), then \(f'(a)=0\) or \(f\) is not differentiable at \(a\)”.
- Write the contrapositive of \(S\).
- Write the converse of \(S\).
- Which of the statements in (a) and (b) are true? If either statement is false, give a counterexample.
Solution
Write \[ P:\ f\text{ has a local minimum at }a, \qquad Q:\ f'(a)=0\text{ or }f\text{ is not differentiable at }a. \] The contrapositive is \(\neg Q\Rightarrow\neg P\), namely:
If \(f\) is differentiable at \(a\) and \(f'(a)\ne0\), then \(f\) does not have a local minimum at \(a\).
This is true because it is logically equivalent to the original statement \(S\).
The converse is \(Q\Rightarrow P\), namely:
If \(f'(a)=0\) or \(f\) is not differentiable at \(a\), then \(f\) has a local minimum at \(a\).
This is false. For example, \(f(x)=x^3\) at \(a=0\) has \(f'(0)=0\) but no local minimum there. Also \(f(x)=x^{1/3}\) is not differentiable at \(0\) and has no local minimum there.