The original question includes a diagram: The graph of y = f'(x): it falls from top left crossing the axis at A, turns at a minimum B below the axis, rises through C on the axis and through D on the positive y-axis, peaks at E, then falls to touch the x-axis at F before rising again.
The function f(x) has derivative f'(x).
The diagram shows the graph of y = f'(x). Coming in from the top left, the curve crosses the x-axis at A, reaches a minimum at B below the axis, rises through C on the x-axis and through D on the positive y-axis, reaches a maximum at E above the axis, then falls to touch the x-axis at F before rising again.
Which point corresponds to a local minimum of f(x)?
- AA
- BB
- CC
- DD
- EE
- FF
Show the answer and worked solution
answer · C
- AA
- BB
- CC
- DD
- EE
- FF
A local minimum of f needs f' to be zero there and to change from negative to positive. On the graph f' is zero at A, C and F. At A it goes from positive to negative, which is a local maximum of f; at F the curve only touches the axis and stays non-negative, so f' does not change sign and f has no turning point there. Only at C does f' pass from negative to positive, so C is the local minimum.