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## Definition Of Increasing Function

A function f is said to be increasing if f(a) > f(b) when a > b.

### More About Increasing Function

The graph of an increasing function always goes up from left to right.
A function is increasing at all the points where its derivative is positive.

### Example of Increasing Function

In figure, the curve is going up from left to right. Hence, the function is said to be increasing.

### Solved Example on Increasing Function

#### Ques: Find the interval in which f(x) = 3x? - 54x2 + 14 increases.

##### Choices:

A. (- 3, 3)
B. (0, 8)
C. (0, 3) U (3, 8)
D. (- 3, 0) U (3, 8)
Correct Answer: D

### Solution:

Step 1: Here f(x) = 3x? - 54x2 + 14 [Given function.]
Step 2: f '(x) = 12x? - 108x [Differentiate.]
Step 3: = 12x (x2 - 9) [Factorize.]
Step 4: = 12x (x - 3) (x + 3) [Factorize.]
Step 5: For x (x + 3)(x - 3) > 0, f ' (x) > 0
Step 6: For x ε (- 3, 0) U (3, 8), f ' (x) > 0 [Solve the inequality.]
Step 7: So, f(x) increases in (- 3, 0) U (3, 8).