# OK. I have a Calc Multiple Choice Problem, but it looks like 2 answers are right?

1. Nov 30, 2011

1. The problem statement, all variables and given/known data
Multiple Choice If f is a continuous, decreasing function on
[0, 10] with a critical point at (4, 2), which of the following statements
must be false? E
(A) f (10) is an absolute minimum of f on [0, 10].
(B) f (4) is neither a relative maximum nor a relative minimum.
(C) f ' (4) does not exist.
(D) f ' (4) = 0
(E) f ' (4) < 0

2. Relevant equations

Ok. It looks to me like C and E are both false, based on the mere fact that D is correct, making this question have 2 answers.
Can someone please explain to me why one of them should be incorrect?

3. The attempt at a solution

2. Nov 30, 2011

### micromass

Staff Emeritus
How did you define critical point??

3. Nov 30, 2011

critical point was where f '(x)= 0, so i said f '(4)=0

4. Nov 30, 2011

### micromass

Staff Emeritus
Under that definition, it seems indeed true that C and E are false. However, I would doublecheck that definition if I were you.

Last edited: Nov 30, 2011
5. Nov 30, 2011

### Ray Vickson

No. A function can be *strictly decreasing* and yet have a critical point. For example, $f(x) = -x^3$ is strictly decreasing but has $f'(0) = 0$. (It is strictly decreasing because for any $x_1 < x_2$ we have $f(x_1) > f(x_2).$)

RGV

6. Nov 30, 2011

### Staff: Mentor

Yeah, critical points aren't just where the derivative is zero...

7. Nov 30, 2011

### micromass

Staff Emeritus
Yes, how does that contradict what I said??

8. Nov 30, 2011

### Ray Vickson

Sorry: it doesn't; I did not read the questions A--E carefully enough.

RGV

9. Nov 30, 2011