Solving for b^2+c^2 with Function and Mapping

[abhip]

The function is one-one find the value of b^2+c^2[f(x)=x^3+3x^2+4x+bsinx+ccosxI tried the approach of monotonocity as a one one function will be strictly inc or decreasing in it's domain but I'm not being able to figure out b^2+c^2

 

[tiny-tim]

welcome to pf!

hi abhip! welcome to pf!

(try using the X² icon just above the Reply box )

hint: what is df/dx ? ​

 

[abhip]

i had tried tat approach of finding df(x)/dx>=0 but can't relate to what has to be found in the question.

 

[tiny-tim]

i had tried tat approach of finding df(x)/dx>=0 …

show us what you got

 

[abhip]

3x^2+6x+4>=csinx-bcosx x belongs to real numbers not how to get a relation with b^2 and C^2 sorry I am finding it difficult using the exponent feature

 

[tiny-tim]

3x^2+6x+4>=csinx-bcosx

now complete the square

 

[abhip]

but how and what to replace the trigonometric expression on RHS of the inequality with
as for LHS it can be written as 3(x+1)^2 +1

 

[tiny-tim]

3(x+1)^2 +1

so the minimum of that is … ?

(oooh, almost forgot … use one of your trigonometric identities to simplify csinx-bcosx )

 

[abhip]

1... but what about b^2 +c^2 how do i introduce them into the inequality because i have a trigonometric expression on the RHS

 

[abhip]

Oh thank you i got it now using polar coordinates...thank you very much...love this forum..it's good that you people are able to solve our doubts it's just impossible getting doubts cleared back at my place as the teachers themselves submit on seeing such doubts...and what i like most is u directly didn't pop the answer it gave me a feeling as if i had been involved in the whole process