Question

State, with explanation, whether the following statements are True or False: a) It is possible for...

State, with explanation, whether the following statements are True or False:

a) It is possible for ? = ?(?) to have an inflection point at (?, ?(?)) even if ?′(?) is not defined.

b) If ?''(?) > 0 then ? has a local minimum at ? = ?.

c) For all functions ?, if ?′(?) exists ∀?, then ?′′(?) exists ∀?.

Math   Calculus   
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Answer #1

(a)

Inflection point can not be defined only if original function is not defined for that x-values

f'(a) or f''(a) may or may not be defined but f(a) must be defined

so, this is TRUE

(b)

f''(c) >0 means concave up

concave up means curve open upward

and function will be opened upward only after local minima

so, this is TRUE

(c)

f''(x) is derivative of f'(x)

derivative only exist when function is continuous

and function is continuous only when function is defined

so, to exist f''(x) , f'(x) must be continuous and differentiable

If f'(x) is continuous but not differentiable then f''(x) does not exist at that point

Hence,

this is FALSE

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