Question

If $f\left(x\right)={\left(\frac{12}{13}\right)}^x+{\left(\frac{5}{13}\right)}^x-1,x\in R$, then the equation $f\left(x\right)=0$ has

• A. no solution
• B. one solution
• C. two solutions
• D. more than two solutions

Answer 1

Answer: (B)

one solution

By inspection, we find that $x=2$ is a solution for the equation $f\left(x\right)=0$.

Let $a=\frac{12}{13},b=\frac{5}{13}$

$\Rightarrow f\left(x\right)=a^x+b^x-1$

Now, $0<a,b<1$

As $x$ increases, $a^x$ and $b^x$ decrease.

Hence for $x>2$ , $f\left(x\right)<0$.

Similarly,

As $x$ decreases, $a^x$ and $b^x$ increase.

Hence, $f\left(x\right)>0$ for $x<2$

Hence, there can be only one solution for $f\left(x\right)=0$