Question

Give another example of a function that increases concave downward on the interval (0, ∞).

Justify your reasoning but do not use a graph.

Answer 1

Example of such function is y = √x

Now, see the justification.

for increasing f'(x) > 0 in the interval

Here f'(x) =

√x is always greater than zero in the given interval.

so,

Now for test for concavity :-

find f"(x) :-

f"(x) = Since x >0 so this expression is always negative.

And if f"(x)<0 then function is concave down.

Another example is

Here f'(x) =

Since x>0 so, 1/x^2 is always greater than O so, f'(x) >0 means function is increasing.

Now, test the concavity:-

​​​​​​f"(x)<0 , Hence concave downward.