Find the points of local maxima or local minima and corresponding loca

1.	Find the points of local maxima or local minima and corresponding local maximum and local minimum values of the functions. Also, find the points of inflection, if any: f(x) = xex
Answer» Given as f(x) = xe^x f'(x) = e^x + x e^x = e^x(x + 1) f”(x) = e^x(x + 1) + e^x = e^x(x + 2) For the maxima and minima, f'(x) = 0 e^x(x + 1) = 0 x = – 1 f’’ (– 1) = e ^{– 1} = 1/e > 0 x = – 1 is point of local minima Thus, local min = f (– 1) = – 1/e

Find the points of local maxima or local minima and corresponding local maximum and local minimum values of the functions. Also, find the points of inflection, if any: f(x) = xex

Answer»

Given as f(x) = xe^x

f'(x) = e^x + x e^x = e^x(x + 1)

f”(x) = e^x(x + 1) + e^x

= e^x(x + 2)

For the maxima and minima,

f'(x) = 0

e^x(x + 1) = 0

x = – 1

f’’ (– 1) = e ^{– 1} = 1/e > 0

x = – 1 is point of local minima

Thus, local min = f (– 1) = – 1/e