# Finding the function from given data

• utkarshakash
In summary, the conversation discusses a problem involving a differentiable function and its properties. The given equation is 6x \displaystyle \int_0^1 f(tx)dt = 2x^3-3x^2+6x+5, and the task is to determine the correct options for the function f(x). The suggested approach is to substitute y = tx and use the Fundamental Theorem of Calculus to differentiate it after transforming the bounds. However, the presence of the constant term (5) on the right-hand side is inconsistent and the question may be wrong.
utkarshakash
Gold Member

## Homework Statement

If f(x) is differentiable function satisfying $6x \displaystyle \int_0^1 f(tx)dt = 2x^3-3x^2+6x+5$ then select correct options(more than one may be correct)

b)f(x)=0 has no real roots
c)f(-x)=f(x+1) for all x in R
d) f(x)=1/2 has 2 real and equal roots

## The Attempt at a Solution

From the options it is clear that the function f(x) is to be determined first. Now, if I divide both sides by 6,
$x \displaystyle \int_0^1 f(tx)dt = \int (x^2 - x +1) dx + \frac{5}{6}$

I have no idea how to take it ahead.

utkarshakash said:

## Homework Statement

If f(x) is differentiable function satisfying $6x \displaystyle \int_0^1 f(tx)dt = 2x^3-3x^2+6x+5$ then select correct options(more than one may be correct)

b)f(x)=0 has no real roots
c)f(-x)=f(x+1) for all x in R
d) f(x)=1/2 has 2 real and equal roots

## The Attempt at a Solution

From the options it is clear that the function f(x) is to be determined first. Now, if I divide both sides by 6,
$x \displaystyle \int_0^1 f(tx)dt = \int (x^2 - x +1) dx + \frac{5}{6}$

I have no idea how to take it ahead.

Hint: In $6x \displaystyle \int_0^1 f(tx)dt$, substitute ##y = tx## and use Fundamental Theorem of Calculus to differentiate it after transforming the bounds.

EDIT: I'm not at all certain that the constant term (5) belongs in the RHS of that equation. It makes no sense to me. If it is disregarded, the question is easily solvable. Maybe someone else will have an insight into this.

Last edited:
Curious3141 said:
Hint: In $6x \displaystyle \int_0^1 f(tx)dt$, substitute ##y = tx## and use Fundamental Theorem of Calculus to differentiate it after transforming the bounds.

A clever approach. Thank You!

utkarshakash said:
A clever approach. Thank You!

I suggest you to make a note of this approach, its going to be helpful a lot of times. Test papers often include problems on FTOC.

The consensus in the homework help forum is that the question is wrong. The presence of the constant term (5) on the RHS is inconsistent. Putting x = 0 makes the LHS vanish but not the RHS.

*Another* wrong question?

Pranav-Arora said:
I suggest you to make a note of this approach, its going to be helpful a lot of times. Test papers often include problems on FTOC.

Thanks for your suggestion. I'm going to make it right now.

## What is the purpose of finding the function from given data?

The purpose of finding the function from given data is to understand the relationship between two or more variables and to be able to make predictions or analyze patterns based on the given data. It is also important for creating mathematical models and understanding real-world phenomena.

## What are the steps involved in finding the function from given data?

The steps involved in finding the function from given data include identifying the independent and dependent variables, plotting the data points on a graph, analyzing the shape of the graph to determine the type of function, and using mathematical techniques such as regression analysis to find the equation of the function that best fits the data.

## What are some common types of functions that can be derived from given data?

Some common types of functions that can be derived from given data include linear functions, quadratic functions, exponential functions, and logarithmic functions. Other types of functions such as trigonometric functions and polynomial functions can also be derived from data in certain situations.

## How accurate are the functions derived from given data?

The accuracy of the function derived from given data depends on the quality and quantity of the data points, as well as the mathematical techniques used to find the function. In general, the more data points and the more accurate the data, the more accurate the derived function will be. However, there may still be some degree of error or uncertainty, especially if the data is noisy or if there are outliers.

## Can a function be derived from any given set of data?

In theory, a function can be derived from any given set of data. However, in practice, the data may not perfectly fit a certain type of function or there may be errors in the data that make it difficult to accurately derive a function. Additionally, some types of functions may not be appropriate for certain types of data. For example, a linear function may not accurately represent data that follows a curved pattern.

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