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# Class 12: All You Want To Know About Implicit function

## Implicit function

We have discussed derivatives of functions of the form y = f(x). If x and y are connected by a relation of the form f(x,y) = 0 and not convenient to write in the form y = f(x), then y is said to be an implicit function of x. in such a case, to find dy/dx , we have to differentiate both sides of the given relation with respect to x.

Remember the following

## Rolle’s Theorem

f(x) is a real function defined in [a,b] having the following properties

1. f(x) is continuous in [a,b]
2. f(x) is differentiable in (a,b)
3. f(a) = f(b)

Then there exist atleast one c between a and b at which fl(c) = 0

1. Verify Rolle’s theorem for the function f(x) = x(x-1) on [0,1]

f(x) is continuous in [0,1], since it is a polynomial function

f(x) id differentiable in (0,1)

f(0) = 0(0-1) =0

f(1) = 1(1-1) = 0

f(0) = f(1)

f(x) = x(x-1)= x² -x

f′(x) = 2x – 1

f′(c) = 2c – 1

f′(c) = 0

2c-1 = 0

2c = 1

c = 1/2

c lies between 0 and 1

Rolle’s theorem is verified

## Mean Value Theorem

f(x) is a real function defined in [a,b] having the following properties

1. f(x) is continuous in [a,b]
2. f(x) is differentiable in (a,b)
3. Then there exist atleast one c between a and b at which
1. Verify mean value theorem for f(x) = x² + 1 in [2,4]

f(x) is continuous in [2,4], since it is a polynomial function

f(x) is differentiable in (2,4)

f(2) = 2² +1 = 5

f(4) = 4² + 1 = 17

f(x) = 2x

f(c) = 2c

f′(c) = [fb-f(a)]/(b-a)

f′(c) = [f(4)-f(2)]/(4-2)

2c = 17-52

2c = 6

C = 3

C lies between 2 and 4

Mean Value Theorem is verified