Question-47

function

maxima and minima

Consider \(\displaystyle f:\mathbb{R}\rightarrow \mathbb{R}\) such that \(\displaystyle f\) is an odd, differentiable function. \(\displaystyle f\) has exactly one local maximum in the interval \(\displaystyle ( 0,\infty )\) which occurs at some positive \(\displaystyle x\). Which of the following statements are true if it is given that \(\displaystyle f^{\prime }( 0) \neq 0\).

\(\displaystyle f\) has exactly one extremum
\(\displaystyle f\) has exactly two extrema
\(\displaystyle f\) has exactly one local minimum
\(\displaystyle f\) has exactly two local minima

Answer

\(\displaystyle f\) has exactly one extremum
\(\displaystyle f\) has exactly two extrema
\(\displaystyle f\) has exactly one local minimum
\(\displaystyle f\) has exactly two local minima

Solution

Since \(\displaystyle f\) is odd, \(\displaystyle f( -x) =-f( x)\). If \(\displaystyle x_{0}\) corresponds to the local maximum in \(\displaystyle [ 0,\infty )\), then \(\displaystyle -x_{0}\) has to be a local minimum. Since \(\displaystyle f^{\prime }( 0) \neq 0\), that cannot correspond to a point of extremum. Since \(\displaystyle f\) is differentiable, we needn’t worry about any other points. Therefore, \(\displaystyle f\) has exactly two extrema, one is a local maximum and the other is a local minimum.