Let $f\left(x\right)$ be differentiable function and $g\left(x\right)$ be twice differentiable function. Zeros of $f\left(x\right),g^{prime}\left(x\right)$ be $a,b$ , respectively, $\left(a<b\right)\dot{}$ Show that there exists at least one root of equation $f^{prime}\left(x\right)g^{prime}\left(x\right)+f\left(x\right)g^x=0$ on $\left(a,b\right)\dot{}$

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