Let f: ℝ → ℝ satisfy f(x + y) = 2x f(y) + 4y f(x), ∀ x, y∈ ℝ. If f(2) = 3, then 14. f'(4)/f'(2) is equal to ___.
Correct Answer: 248
Solution and Explanation
The correct answer is 248
∵ f(x + y) = 2x f(y) + 4y f(x) …(1)
Now, f(y + x) 2y f(x) + 4x f(y) …(2)
∴ 2x f(y) + 4y f(x) = 2y f(x) + 4x f(y)
(4y – 2y) f(x) = (4x – 2x) f(y)
\(\frac{ƒ(x)}{4x - 2x} = \frac{ƒ(y)}{4y - 2y} = k (Say)\)
∴ f(x) = k(4x – 2x)
∵ f(2) = 3 then
\(k = \frac{1}{4}\)
\(∴ ƒ(x) = \frac{4x - 2x}{4}\)
\(∴ ƒ′(x) = \frac{4^xIn4 - 2^xIn2}{4}\)
\(ƒ′(x) = \frac{(2.4^x - 2^x ) In2}{4}\)
∴ \(\frac{ƒ′(4)}{ƒ′(2)}= \frac{2.256 - 16}{2.16 - 4}\)
\(∴ 14 \frac{ƒ′(4)}{ƒ′(2)} = 248\)
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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.
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