Question

find the value of x for which
( 1 )
$f(x) = x {3}^{} \: - 6x {}^{2} + 12x \: + 10 \: is \: increasing$
is increasing.

( 2 )
$f(x) = 3x {}^{2} - 15x + 9 \: is \: decreasing$
​

Answer 1

Answer:

f(x)=2x

3

−3x

2

−12x+6

∴f

′

(x)=

dx

d

(2x

3

−3x

2

−12x+6)

=2×3x

2

−3×2x−12×1+0

6x

2

−6x−12

6(x

2

−x−2)

f is strictly increasing if f

′

(x)>0

i.e. if 6(x

2

−x−2)>0

i.e. if x

2

−x−2>0

i.e. if x

2

−x>2

i.e. if x

2

−x+

4

1

>2+

4

1

i.e. if (x−

2

1

2

>

4

9

)

i.e. if x−

2

1

>

2

3

or x−

2

1

<−

2

3

i.e. if x>2 or x<−1

∴f is strictly increasing if x<−1 or x>2.