Find the relative maxima and relative minima, and sketch the graph with a graphing calculator to check your results. (If an answer does not exist, enter DNE.)

Question

Find the relative maxima and relative minima, and sketch the graph with a graphing calculator to check your results. (If an answer does not exist, enter DNE.)

y = 9x ln(x)

relative maximum (x,y) = ( ? )

relative minimum (x,y) = ( ? )

Answer 1

First need to take the derivative of y, set that to zero and solve for x.

y' = 9[ln(x) +x(1/x)] = 9ln(x) + 9

9ln(x) = -9

ln(x) = -1

x = e^-1 = 1/e

y(e^-1) = 9e^-1ln(e^-1) = -9/e

(x,y) = (1/e, -9/e)

Do the 2nd derivative test

y"(x) = 9/x

y"(1/e) = 9e > 0 so (1/e, -9/e) is a relative minimum

relative maximum DNE

relative minimum (x,y) = (1/e, -9/e)