Question

Number of real solution of the equation sin (e^x) = 5^x + 5^-x is

Answer 1

Correct Answer - Option 1 : 0

Concept:

-1 ≤ sin x ≤ 1

 

AM, GM, HM Formulas

If A is the arithmetic mean of numbers a and b and is given by ⇔ \({\rm{A}} = \frac{{{\rm{a\;}} + {\rm{\;b}}}}{2}\)

If G is the geometric mean of the numbers a and b and is given by ⇔ \({\rm{G}} = \sqrt {{\rm{ab}}} \)

If H is the Harmonic mean of numbers a and b and is given by ⇔ \({\rm{H}} = \frac{{2{\rm{ab}}}}{{{\rm{a}} + {\rm{b}}}}\)

Relation between AM, GM and HM

1. G2 = AH
2. AM  ≥  GM  ≥  HM

 

Calculations:

Given, the equation is sin (ex) = 5x + 5-x 

Consider, LHS = sin (ex)  < 1

As we know AM ≥ GM

\(\rm \Rightarrow \frac{5^x + 5^{-x}}{2} \geq (5^x \times 5^{-x})^{1/2}\)

∴ 5x + 5-x  ≥ 2

RHS = 5x + 5-x  ≥ 2

Here, LHS \(\neq\) RHS

⇒The equations sin (ex) = 5x + 5-x have no solution. This means that no matter what value is plugged in for the variable, you will ALWAYS get a contradiction.

Hence, Number of real solution of the equation sin (ex) = 5x + 5-x is zero.