f is a twice differentiable function that is defined for all reals. The value of f "(x) is given for several values of x in the table below.

Question

f is a twice differentiable function that is defined for all reals. The value of f "(x) is given for several values of x in the table below.

x–8–3038
f " (x)	–4	–2	0	4	5

If f " (x) is always increasing, which statement about f(x) must be true? 



f(x) is concave downwards for all x. 

f(x) passes through the origin. 

f(x) has a relative minimum at x = 0.

 f(x) has a point of inflection at x = 0.

Answer 1

When the 2nd derivative [ f '''(x) ] is NEGATIVE, it means the graph of the original f(x) is concave down.

When the 2nd derivative [ f ''(x) ] is POSITIVE, it means the graph of the origninal f(x) is concave up.

When the 2nd derivative [ f ''(x) ] = 0, it means it is neither concave up nor concave down, but possible switching from one to the other, this is called a point of inflection.

YOUR f '' is neg, zero, and pos at different x-values, therefore the original functions would be conc down at the beginning, then at the point of inflection (where f ''(x) = 0) it switches to concave up (as seen by the positive f ''(x)'s)

So the original graph either looks like y=x³ or y = -x³ (a "snake" I tell my students - wish I could draw here...)

f(x) is concave downwards for all x. FALSE

f(x) passes through the origin. (cannot tell from this data)

f(x) has a relative minimum at x = 0. FALSE

 f(x) has a point of inflection at x = 0. TRUE

Last statement is the correct one.