Minima or Maxima

Question

Approximate (to two decimal places) any relative minima or maxima

1. f(x) = 3x² - 2x - 5
2. f(x) = x³ - 3x² - x + 1
3. f(x) = x(x-2)(x+3)

Answer 1

extrema occur when first derivative (slope of the tangent line) is zero

1) y = 3x^2 - 2x - 5

0= y' = 6x - 2

x = 1/3

2) y = x^3 - 3x^2 - x + 1

0 = y' = 3x^2 - 6x - 1

x = [6 +or- sqrt( 36 + 12)]/6

= [6 +or- sqrt(48)]/6

= [6 +or- 4*sqrt(3)]/6

= [3 +or- 2*sqrt(3)]/3

3) y = x(x-2)(x+3)

y = x(x^2 + x - 6)

y = x^3 + x^2 - 6x

y' = 3x^2 + 2x - 6

x = [-2 +or- sqrt( 4 + 72)]/6

= [-2 +or- sqrt(76)]/6

= [-2 +or- 2*sqrt(19)]/6

= [-1 +or- sqrt(19)]/3