Question

For a function defined on an interval \[I\], which condition means that \[f\] has a minimum value at \[c\in I\]?

Options

• \[f'(c)<0\quad\text{for all }x\in I\]

• \[f(c)\leq f(x)\quad\text{for all }x\in I\]

• \[f''(c)>0\quad\text{for all }x\in I\]

• \[f(c)\geq f(x)\quad\text{for all }x\in I\]

Answer 1

A minimum value at \[c\] is no greater than every function value on \[I\]. Hence \[f(c)\leq f(x)\] for all \[x\in I\].