Maximizing area under $y=e^{−{∣x∣}}$ The coordinates of the point $M(x,y)$ on $y=e^{−{∣x∣}}$ so that the area formed by the coordinates axes and the tangent at $M$ is greatest is what?
I tried to plot the graph but after that I'm not being able to proceed.Please help]1

the problem is symmetric so will look at the tangent at $P=(1, 1/e^a), a > 0$ and the area of the region bounded by the tangent, positive $x$-axis and the positive $y$-axis.
the tangent at  $P$ has slope $-\frac1{e^a}$ and is given by $$y - \frac1{e^a}=-\frac1{e^a}\left( x - a \right) $$ the $x$-intercept is $(1+a, 0)$ and the $y$-intercept is $\frac{1+a}{e^a}.$ so the required area is $$A = \frac{(1+a)^2}{2e^a}, 0 \le a < \infty \tag 1$$ 
you can verify that $A$ has a global max of $\frac 2 e$ at $a = 1.$   
