Graph the function below over the indicated interval hx x3x

Graph the function below over the indicated interval. h(x) = x(3^x); [-4.0] Choose the correct graph below.

Solution

On the given interval: x<=0 and 3^x>0

So, h(x)<=0

At x=0,h(0)=0

As magnitude of x increases 3^x goes to 0 and since 3^x goes to 0 faster than x goes to -infinity the whole graph goes to 0

A,B,C seem to fit this description.

So we now look at second derivatives to find curvature of graph as x goes to -infinity

h\'=3^x+x3^x log(3)

h\'\'=2log(3) 3^x + x3^x log(3)^2=3^x log(3)(2+x log (3))

So we see there is inflection point at:x=-2/log(3) hence, A and C seem to fit the description.

the critical point ie where derivative is 0 is x=-1/log(3)

At this point the value of function is

h(-log(3))~-0.3

Hence, C is correct option