Question

Determine whether each of the following statements is true or false, and explain why in a few sentences

1. A critical number c is a number in the domain of a function f for which f ′ ( c ) = 0 or f ′ ( c ) does not exist.

2. If f ′ ( c ) > 0 on an interval, the function is positive on that interval.

3. If c is a critical number, then the function must have a relative maximum or minimum at c

4. If f ′ ( c ) exists, then f ″ ( c ) also exists

5. If f ″ ( c ) > 0 on an interval, the function is increasing on that interval.

Answer 1

1. TRUE Because the definition of Critical point (number is "A Critical number of a function of is a number ein the domain off such that either f(e)=0 or f'(c) does not exist. 2. ARNE FALSE Because if f'(c) >0 on an interval, then fis. increasing on that interval Can't Say It is positive. If fwy = én it is negative - but f'G) = e "20 xhe Dan (0%) Take flor=v3 on (-2,2] f'lar= 322 f (C) = 3 c² so v c E-22] But fl-17 =-1 is not positive So Statement is falte 3. (FALSE Take f(x)= x3 has a Costical number at aro yet flo) is neither a local (relative Maximum nos a relative minimum.

4. FALSE If fon2= x2 362 then f'G) = 3 x 2 = 3 at u=0 f'(o)=0 Exists. f"(W) = 3 1 an afn=o f'(o) Desnot exist at all 5. FALSE) If f(n)=12 on [-2,23 then f'G11=24 f(12= 2 >0 & NEE22] at u=CE[22] f"CC2=2>0 FC E [22] but food = 82 is decreasing in [20] given Statement is false
I've solved these questions for you, kindly let me know if you have any doubts in it.