If \(f(x) - f(y) = ln\bigg(\frac{x}{y}\bigg) +x-y\), then find \(\sum^{20}_{k=1}f'\bigg(\frac{1}{k^2}\bigg)\)
- 2890
- 2390
- 1245
- None of this
The Correct Option is A
Solution and Explanation
The correct option is (A): \(2890\)
\(f(x)-In(x)-x=f(y)-In\, y-y\)
\(=f(x)-In(x)-x=c\)
\(=f(x)=c+x+In \,x\)
f"(x)=0+1+\(\frac{1}{x}\)
After simplification we get:
\(20+\frac{20\times21\times41}{6}\)
\(20+2870=2890\)
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Concepts Used:
Functions
A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Let A & B be any two non-empty sets, mapping from A to B will be a function only when every element in set A has one end only one image in set B.
Kinds of Functions
The different types of functions are -
One to One Function: When elements of set A have a separate component of set B, we can determine that it is a one-to-one function. Besides, you can also call it injective.
Many to One Function: As the name suggests, here more than two elements in set A are mapped with one element in set B.
Moreover, if it happens that all the elements in set B have pre-images in set A, it is called an onto function or surjective function.
Also, if a function is both one-to-one and onto function, it is known as a bijective. This means, that all the elements of A are mapped with separate elements in B, and A holds a pre-image of elements of B.
Read More: Relations and Functions