Mathematics - Application of Derivative Question with Solution | TestHub

Slope of the line joining the points $\left(c-1,e^{c-1}\right)$ and $\left(c+1,e^{c+1}\right)$ is equal to $\frac{e^{c+1}-e^{c-1}}{2}>e^c$
$$\text{ Tangent to the curve }y=e^x\text{ will intersect the given line to the left of the line }x=c\text{. }$$ ALITER The equation of the tangent to the curve $y=e^x$ at $\left(c,e^c\right)$ is $$y-e^c=e^c(x-c)$$ Equation of the line joining the given points is $$y-e^{c-1}=\frac{e^c\left(e-e^{-1}\right)}{2}[x-(c-1)]$$ Eliminating $y$ from Eqs. (i) and (ii), we get $$\begin{array}{rl}[x-(c-1)]\left[2-\left(e-e^{-1}\right)\right]& =2e^{-1}\\ x-c& =\frac{e+e^{-1}-2}{2-\left(e-e^{-1}\right)}<0\Rightarrow x<c.\end{array}$$