Question

for what values of x is the graph concave up? -3e^(-3x)+2x(e^(-3x))

Answer 1

\[-3e ^{-3x}+2xe ^{-3x}\]

Answer 2

find the 1st and 2nd derivatives

let the 1st derivative = 0 and solve for x

this will give stationary points

then find the 2nd derivative

solve the 2nd derivative for x this will give possible points of inflection.

test the stationary points in the 2nd derivative for concavity

test either side of the the 2nd derivative solution for a change in concavity.

if you have a point of infection the curve will be concave up to the right of the point of inflection...