Question

Determine the intervals on which the graph of ? = ?(?) is concave up or concave down, and find the ?-values at which the points of inflection occur.

?(?)=? (? − 9 √x), ? > 0

(Enter an exact answer. Use symbolic notation and fractions where needed. Give your answer in the form of a comma separated list, if necessary. Enter DNE if there are no points of inflection.)

? =

(Use symbolic notation and fractions where needed. Give your answers as intervals in the form (∗,∗).

Use the symbol ∞ for infinity, ∪ for combining intervals, and an appropriate type of parenthesis "(", ")", "[", or "]", depending on whether the interval is open or closed. Enter ∅

if the interval is empty.)

? is concave up when ?∈

? is concave down when ?∈

Answer 1

we are given function as

we can find second derivative

we can find derivative again

now, we can set it to 0

and solve for x

now, we can draw a number line and locate 11.390

and then we can check sign of second derivative on each intervals

Concave down intervals:

Concave up interval:

Inflection point:

Since, concavity changes at x=11.390

so, there will be inflection point at x=11.390