Consider a function f on the interval [ -3,4 ] with the following properties:
- f is continuous on [ -3,4 ]
- f(0) = 0
- f'(1) = 0 and f'(3) = 0
- f'(x) = -1 on the interval (-3,-1)
- f'(x) > 0 and f''(x) < 0 on the interval (-1,0)
- f'(x) > 0 on the interval (1,3)
- f is concave down on (2,4) and concave up on (0,2)
(a) determine ( also explain ) whether the following statements are true or false.
- Argue briefly that f must have a local minimum at x = -1
- True or false. the absolute minimum of f is attained somewhere on [-3,4]
- Is there an inflection point at x = 0 ? why or why not?
(b) sketch an example of a function f with the above properties.