IIT JEE , AIEEE Forum - Mathematics , Algebra

faith

Consider this eqn

4x⁴+ 5 x³+ 7x² + 2x = 0

diff. once , twice, thrice resp.

this implies 16 x^3 + 15 x^2 + 14 x + 2=0

this implies 48 x^2 + 30 x + 14 = 0

this implies 96 x + 30 = 0

ie x= -30/96

which doesnt satisfy the original eqn, although is derived from the same

utkarsh123

rate me then i will tell

nitin62225

it doesn't necessarily mean that derivative is zero then the function possesses a root at that point!!!

it happens in cases where u have repeated roots(equal roots)

swaroopcool21

when one takes the first derivative = 0

it implies that function has minima/maxima at that point....

but u can't just diffrentiate the equation ............

as derivative is actually the slope of that curve

so doesnt necessarily satisfy the given equation...

faith

first of all i wanna clear it out dat i didnt give a function, but an equation.

and 2ndly when u get something form the previous step, it shud satisfy the previous step , otherwise its wrong., isnt it?

magiclko

its not necessary...

see here's a simple example

in the attached figure, thr's a graph of a parabola...which is a function...

if we differntiate the function we'll be getting f ' (h) = 0, while f(h) = k

0

same is the case for ur question... it isnt at all necessary..what u r saying

edison

Q)

Consider this eqn

4x⁴+ 5 x³+ 7x² + 2x = 0

diff. once , twice, thrice resp.

this implies 16 x^3 + 15 x^2 + 14 x + 2=0

this implies 48 x^2 + 30 x + 14 = 0

this implies 96 x + 30 = 0

ie x= -30/96

which doesnt satisfy the original eqn, although is derived from the same

Ans) Infact differentiating an equation to find its solution is wrong method and will never work.

Infact differential equations in themselves represent FAMILY OF EQUATIONS and not simply one equation.

For example in your case you startewith

4x⁴+ 5 x³+ 7x² + 2x = 0

similarly if you add some constant to this equation then we may have

4x⁴+ 5 x³+ 7x² + 2x + 5= 0

4x⁴+ 5 x³+ 7x² + 2x + 20 = 0

4x⁴+ 5 x³+ 7x² + 2x + 100 = 0

now all above three equations differ by a constant and if you proceed in a similar fashion you reach at

x = -30/96

Then will you expect that the single solution will satisfy all above three equations.

OBVIOUSLY NOT

(because differentiting equation gives family of several such equations that may differ by simply any arbitrary constant)

I suppose things are now clear to you.

iitkgp_bipin

Actually it can't be derived from the same equation.

Differentiation can be applied to both the sides of the equation if it is valid for all values in domain.

The equation 4x⁴+ 5 x³+ 7x² + 2x = 0 is not valid for any x as it can have maximum of 4 roots.

Actually the above equation represents the intersection of the curves y=4x⁴+5 x³+7x²+2x and y=0 and it is not the intersection of their slopes.

amar.gupta

Dear faith,

I think you have got the point as every body has tried and given almost all types of possible explanation to explain your query.

The derivative is the instantaneous rate of change of a function with respect to one of its variables. This is equivalent to finding the slope of the tangent line to the function at a point.

as in physics , you might have done that a=dv/dt i.e. rate of change of velocity is the acceleration.

so it shows the behavior of the eq. or function whatever you say .
it shows where the function is max. or min. for which value of x but it does not give the values of minimum and maximum . you have to get the value by putting those values in real eq..

so now i think you get the point .

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