Question

Let f be a twice differentiable function on an open interval (a, b). 

Which statements regarding the second derivative and concavity are true? 

• If f"(c) is positive, then the graph of f has a local maximum at x = c. 

• The concavity of a graph changes at an inflection point. 

• If f is increasing, then the graph of f is concave down. 

• The graph of f has a local minimum at x = c if f"(c) = 0. 

• The graph of f is concave up if f" is positive on (a, b).

Answer 1

Given is twice differentiable on (a,b) (i) Take a Gi) example ob f(x)= x2 f(!)=2x f!!C70 = 2 these f'cc) is positie But it is not local maximum. so option is is not correct. Inflection point is point, where of!" () =0 and know that f(u)20 and f(") Lo de fine concavity, so at inflection point graph changes conearity. we (ii) This is increasing graph but it is not Concave down, so option (ii) is not correct (v) point where f'(o)zo we get inflection point not point of minima and maxima, so option (iv) also not correct. ♡ Yes it f (x) >0 for (a,b) then for this interval concave up f (H) will be option () is correct. So, option (ii) and (y) are correct.

Answer 2

ANSWER :

Option 1 : Not True ,since if f” (c) is positive then it may have a minimum at point (c) but not maximum.

Option 2 : True,  since at point of inflection, concavity of the function changes.

Option 3 : Not True, as If f is increasing it may be either concave up or concave down. Both possibilities are there.

Option 4 : Not True, since for local maximum or minimum  at f(c), 

f ‘ (c) should be equal to zero. f “ (c) may be > 0 or < 0 or = 0.

Option 5 : True,  since if f “ is positive , f ‘ is increasing which happens in concave up graph of f.

Option 2 and 5 are TRUE . (ANSWER).