Book: RD Sharma - Mathematics (Volume 1)

Chapter: 17. Increasing and Decreasing Functions

Subject: Maths - Class 12th

Q. No. 22 of Exercise 17.2

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22
State when a function f(x) is said to be increasing on an interval [a, b]. Test whether the function f(x) = x2 – 6x + 3 is increasing on the interval [4, 6].

Given:- Function f(x) = f(x) = x2 – 6x + 3

Theorem:- Let f be a differentiable real function defined on an open interval (a,b).

(i) If f’(x) > 0 for all , then f(x) is increasing on (a, b)

(ii) If f’(x) < 0 for all , then f(x) is decreasing on (a, b)

Algorithm:-

(i) Obtain the function and put it equal to f(x)

(ii) Find f’(x)

(iii) Put f’(x) > 0 and solve this inequation.

For the value of x obtained in (ii) f(x) is increasing and for remaining points in its domain it is decreasing.

Here we have,

f(x) = f(x) = x2 – 6x + 3

f’(x) = 2x – 6

f’(x) = 2(x – 3)

Here A function is said to be increasing on [a,b] if f(x) > 0

as given

x ϵ [4, 6]

4 ≤ x ≤ 6

1 ≤ (x–3) ≤ 3

(x – 3) > 0

2(x – 3) > 0

f’(x) > 0

Hence, condition for f(x) to be increasing

Thus f(x) is increasing on interval x [4, 6]

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