Answer
The function $f(x)$ is continuous on $[0 ,1$ with $f(0)=1 \gt 0$ and $f(2)=-3 \lt 0$
Therefore, by the Intermediate Value Theorem, there is a $c \in (0, 2)$ such that $f(c)= 2^c+3^c -4^c =0 \implies 2^c+3^c =4^c$. So, the equation $2^x +3^x=-4^x $ has a solution $c$ in $(0,2)$.
Work Step by Step
We are given that $f(x)=2^x +3^x -4^x$
The function $f(x)$ is continuous on $[0 ,1$ with $f(0)=1 \gt 0$ and $f(2)=-3 \lt 0$
Therefore, by the Intermediate Value Theorem, there is a $c \in (0, 2)$ such that $f(c)= 2^c+3^c -4^c =0 \implies 2^c+3^c =4^c$. So, the equation $2^x +3^x=-4^x $ has a solution $c$ in $(0,2)$.
