Fall 2019

Total marks: 10

Due Date: 30th December, 2019

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Question:

If Find the intervals where the given function is

- Increasing
- Decreasing
- Concave up
- Concave down respectively.

Also find the point of inflection for the given function.

Solution:

```
Increasing
```

f(x)=x3-27x

After Taking Derivative

f `(x)=3x2-27

Function will increase if it is greater than zero

f `(x)=3x2-27>0

3x2-27>0

x2-9>0

x2>9

After taking square root

x>3

So, function increase on (3,∞)

```
Decreasing
```

f(x)=x3-27x

Take Derivative

f `(x)=3x2-27

Function will decrease when it is less than zero

f `(x)=3x2-27<0

3x2-27<0

x2-9<0

x2<9

After taking square root

x<3

So, function decrease on (- ∞, 3)

```
Concave up
```

f(x)=x3-27x

After Taking Derivative

f `(x)=3x2-27

After taking second derivative

f ``(x)=6x

f ``(x)=6x>0

6x>0

x>0

So, it is concave up on (0,+∞)

```
Concave down
```

f(x)=x3-27x

After Taking Derivative

f `(x)=3x2-27

After taking second derivative

f ``(x)=6x

f ``(x)=6x<0

6x<0

x<0

So, it is concave down on (- ∞,0)

POINT OF INFLECTION

f(x)=x3-27x ------------------ (1)

After Taking Derivative

f `(x)=3x2-27

After taking second derivative

f ``(x)=6x

f ``(x)= 6x =0

6x=0

x=0

Put this in (1)

f(0)=(0)3- 27(0)

f(0)=0

So, inflection point is (0 , 0)

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