Find the critical numbers of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) g(x) = e^12x(sinh(8x)) x =

Question

Find the critical numbers of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

g(x) = e^12xsinh(8x)

x =

Answer 1

g'(x) = 12e^12xsinh(8x) + 8e^12xcosh(8x) = 0; e^12x ≠ 0, so 3sinh(8x) + 2cosh(8x) = 0

tan(8x) = - 2/3; (e^8x - e^-8x)/(e^8x + e^-8x) = - 2/3; 3e^8x - 3e^-8x = - 2e^8x - 2e^-8x; 5e^8x = e^-8x; e^-16x = 5;

x = - ln5/16 = - 0.10059