find all of the critical points and local maximums and minim

find all of the critical points and local maximums and minimums of each

Solution

for critical point f\'(x) = 0,

1) f(x) = x² + 8x + 7

f\'(x) = 2x + 8 = 0 => x = -4 (critical point)

f\'\'(x) = 2 >0 (-4 is point of minima)

local minimum = f(-4) = -9

2) f(x) = x³ - 6x² + 5

f\'(x) = 3x² -12x = 0 => x =0, x = 4

f\'\'(x) = 6x - 12,

f\'\'(0) = -12 (maxima) => f(0) = 5

f\'\'(4) = 12 (minima) => f(4) = -27

3) f(x) = ln(x² - 6x + 11)

f\'(x) = (2x- 6)/(x² - 6x + 11) = 0 => x =3

f\'\'(x) = [2(x² - 6x + 11) - (2x -6)(2x -6)]/(x² - 6x + 11)²

f\'\'(3) = 1 (minima) f(3) = ln2