Question

The number of real solutions of the equation $sin\left(e^x\right)=5^x+5^{-x}$is(are)

• A. $0$
• B. $1$
• C. $2$
• D. infinitely many

Answer 1

The correct option is A $0$

Let $5^x=a^2$

$a^2+\frac{1}{a^2}$

$a^2+\frac{1}{a^2}-2+2$
${(a-\frac{1}{a})}^2+2$

${(a-\frac{1}{a})}^2+2\ge 2$

Now $AM\ge GM$

$\frac{5^x+5^{-x}}{2}\ge (5^x{.5}^{-x}{)}^{1/2}$
$5^x+5^{-x}\ge 2$
$-1\le sin\left(e^x\right)\le 1$

$sin\left(e^x\right)\ne 5^x+5^{-x}$