The number of real solutions of the equation e4x + 4e3x - 58e2x + 4ex + 1 = 0 is _____.
Correct Answer: 2
Solution and Explanation
The given equation e4x + 4e3x - 58e2x + 4ex + 1 = 0
Dividing by e2x
e2x + 4ex – 58 + 4e–x + e–2x = 0
⇒ (ex + e–x)2 + 4(ex + e–x) – 60 = 0
Let ex + e–x = t ∈ [2, ∞)
⇒ t2 + 4t – 60 = 0
⇒ t = 6 is only possible solution
ex + e–x = 6
⇒ e2x – 6ex + 1 = 0
Let ex = p,
p2 – 6p + 1 = 0
\(⇒ p = \frac {3+\sqrt 5}{2}\) or, \(\frac {3-\sqrt 5}{2}\)
Therefore,
\(x = ln (\frac {3+\sqrt 5}{2})\)
or, \(x=ln (\frac {3-\sqrt 5}{2} )\)
So, The number of real solutions of the equation is 2.
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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