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 each of the following functions. Also, find the points of inflection, if any : f(x) = x3 – 6x2 + 9x + 15
Answer» f(x) = x³ – 6x² + 9x + 15 ∴ f'(x) = 3x² – 12x + 9 = 3(x² – 4x + 3) f''(x) = 6x – 12 = 6(x – 2) For maxima and minima, f'(x) = 0 3(x² – 4x + 3) = 0 So roots will be x = 3, 1 Now, f''(3) = 6 > 0 x = 3 is point of local minima f''(1) = – 6 < 0 x = 1 is point of local maxima local max value = f(1) = 19 local min value = f(3) = 15

Find the points of local maxima or local minima and corresponding local maximum and local minimum values of each of the following functions. Also, find the points of inflection, if any : f(x) = x3 – 6x2 + 9x + 15

Answer»

f(x) = x³ – 6x² + 9x + 15

∴ f'(x) = 3x² – 12x + 9

= 3(x² – 4x + 3)

f''(x) = 6x – 12

= 6(x – 2)

For maxima and minima,

f'(x) = 0

3(x² – 4x + 3) = 0

So roots will be x = 3, 1

Now,

f''(3) = 6 > 0

x = 3 is point of local minima

f''(1) = – 6 < 0

x = 1 is point of local maxima

local max value = f(1) = 19

local min value = f(3) = 15