Mathematics - Application of Derivative Question with Solution | TestHub

Clearly, $(1,0)$ is the point of intersection of the given curves. Now, $f^{\prime }(x)=\frac{2^x}{x}+2^x\left({\log }_{e}2\right)\left({\log }_{e}x\right).\therefore$ slope of tangent to the curve $f(x)$ at $(1,0),m_1=2$. $g^{\prime }(x)=\frac{d}{dx}\left(e^{2x\log x}-1\right)=x^{2x}\left(2x\times \frac{1}{x}+2{\log }_{e}x\right).\therefore$ slope of tangent to the curve $g(x)$ at $(1,0),m_2=2$. Since $m_1=m_2=2$, two curves touch each other, so the angle between them is 0 . Hence, $\cos \theta =\cos 0=1$.