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If f(x)=x 3 +3x 2 +12x-2sinx,where f is a function from R to R,then 1.f(x) is many-one and onto 2.f(x) is one-one and onto 3.f(x) is one-one and into 4.f(x) is many-one and into

If f(x)=x3+3x2+12x-2sinx,where f is a function from R to R,then


1.f(x) is many-one and onto


2.f(x) is one-one and onto


3.f(x) is one-one and into


4.f(x) is many-one and into

Grade:12th Pass

2 Answers

Rahul Saxena
14 Points
10 years ago

f(x) is many one and onto

Ashwin Muralidharan IIT Madras
290 Points
10 years ago

Hi Meanks,

 

Solving this question, we would take the Differential Calculus approach.

 

Let y = f(x)

Then we have dy/dx = 3x^2 + 6x + 12 - 2cosx

Now, take a look at 3x^2 + 6x + 12, the minimum value of which is 9, when x = -1. And hence dy/dx is always positive (since cosx can take max value of 1, so dy/dx should always be positive)

Which will tell us that y = f(x) is an increasing function in its entire Domain R.

Which would give us that f is one-one function (as continuously increasing functions are always one-one because a line drawn parallel to the y-axis can intersect the graph in only one point)

 

Next also clearly f(x) is continuos, and also f(x) would tend to infinity when x tends to infinity, and f(x) would tend to -infinity when x tends to -infinity.

So the graph would be something like this:

And so f(x) is both one-one and onto. Which is option (2).

 

Hope that hepls,

 

All the best,

Regards,

Ashwin (IIT Madras)

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