Which of the following statement(s) is/are true?

Options

A. a) Differentiable function satisfying `f(-1)=f(1)` and `f'(x)ge0` for all x must be a constant function on the interval `[-1,1]`.
B. b) There exists a function with domain R satisfying f(x) lt 0, for all x, f'(x) gt 0 for all x and f''(x0 gt 0 for all x.
C. c) If f''(x) = 0 then (c,f(c)) is an inflection point.
D. d) Suppose f(x) is a function whose derivative is the function `f(x)=2x^(2)+2x-12`. Then f(x) is decreasing for `-3 lt xlt 2` and concave up for ` xgt-(1)/(2).`

Answer

A, D

Explanation

(a) True
If f(x) is not constant on `[-1,1]`
`rArr" "f` is increasing at that point and we can't have
`f(-1)=f(1)`
(b) Not true
If f is bounded curve changes its concavity
If f is unbounded then curve crosses x-axis
(c) Not true
e.g.f(x)=`x^(4)`
(d) `f'(x)=2x^(2)+2x-12=2(x+3)(x-2)lt0` for `x in (-3,2)`
where f(x) decreases.