1. Find the turning points of each of the following functions and determine their nature by examining the gradient on each side of the turning point.

a
f(x) = 6x − x2
b
f(x) = 8x + x2
c
y = x3
d
y = 1 − 3x + 3x2 − x3
e
g(x) = x3 − 12x
f
y = 27x − x3
g
f(x) = √(9 − x2)
h
y = 
i
f(x) =
j
f(x) = 
k
y = 192x − 88x2 + 16x3 − x4
l
y = (x − 3)4
m
f(x) = (x + 2)3
n
g(x) = x4 − 8x3 + 18x2
o
y = x(x − 6)2

2. Find the turning points of each of the following functions and determine their nature by using the second derivative test.

a
f(x) = 8x − x2
b
y = 5x + x2
c
g(x) = x3 + 3x2 − 105x
d
y = 48x − x3
e
y = x3 − 27x + 1
f
f(x) = 12x2 − x3
g
y = 12x − x3
h
f(x) = 6x2 − 2x3
i
g(x) = x3 + 3x2 + 3x + 1
j
f(x) = x4
k
y = x4 − 2x3
l
f(x) = ex
m
y = x(x2 + 3)
n
f(x) = 4x3 − x5
o
y = x4 + 4x3 − 2x2 − 12x