Let’s now look at a few examples of increasing and decreasing curves

- Find the value of x for which the curve y = x
^{3}– 3x^{2}+ 27x – 3 is increasing.

y = f(x) = x^{3}– 3x^{2}+ 27x – 3

y’ = f'(x) = 3x^{2}– 6x + 27= 3(x

^{2}– 2x + 9)For the function to be increasing f'(x) > 0

So x

^{2}– 2x + 9 > 0This is a quadratic function, and has a >0

b

^{2}– 4ac = 4 – 36 = -32 (which is < 0)Hence f'(x) i.e. x

^{2}– 2x + 9 does not have any real solutions, and the parabola does not meet x-axis. It is above x-axis and always increasing.

**Hence this is a monotonic increasing curve.** - Show that y = 2x
^{3}– 12x^{2}+ 24x – 16 is a monotonic curve and also find its stationary point.

y = 2x^{3}– 12x^{2}+ 24x – 16

y’ = 6x^{2}– 24x + 24= 6(x

^{2}– 4x + 4)= 6(x – 2)

^{2}(x – 2)

^{2}> 0 for all x, so x is always positiveTherefore y’ > 0 for all x.

**Hence the curve is monotonic increasing**.For stationary point –

f'(x) = 0

6(x – 2)

^{2}= 0(x – 2) = 0

So x = 2.

At x = 2, y = 2(2)

^{3}– 12(2)^{2}+ 24×2 – 16 = 0So (2, 0) is the stationary point.