FORM GR1268 Question 9

Question:

Let \(g\) be a continuous real-valued function defined on \(\mathbb R\) with the following properties.

\[g'(0)=0\]

\[g^{\prime\prime}(-1)\gt0\]

\[g^{\prime\prime}(x)\lt0\text{   if   }0\lt x\lt 2.\]

Which of the following could be part of the graph of \(g\)?

Answer:

(A)

Answer Key:

Since \(g'(0)=0\), the slope of \(g\) at \(x=0\) is zero.

Since \(g^{\prime\prime}(-1)\gt0\), \(g\) is convex at \(x=-1\).

Since \(g^{\prime\prime}(x)\lt0\) for \(0\lt x\lt 2\), \(g\) is concave over \(0\lt x\lt 2\).

Think about this.

When \(g^{\prime\prime}\gt0\):

For \(g'\lt0\), the slope is getting less negative.

For \(g'\gt0\), the slope is getting more positive (steeper).

So, \(g\) is convex.

When \(g^{\prime\prime}\lt0\):

For \(g'\gt0\), the slope is decreasing (less steep).

For \(g'\lt0\), the slope is getting more negative.

So, \(g\) is concave.

Or, you can use memory trick like \(+\,\,_\cup+\) for convexity and \(-\,\,_\cap-\) for concavity.