Question

What are the local extrema of `f(x)= 1/sqrt(x^2+e^x)-xe^x`?

Answer 1

By graphical method, local maximum is 1.365, nearly, at the turning point (-0.555, 1.364), nearly. The curve has an asymptote `y = 0 larr`, the x-axis.

Explanation:

The approximations to the turning point (-0.555, 1.364), were obtained by moving lines parallel to the axes to meet at the zenith.

As indicated in the graph, it can be proved that, as `x to -oo, y to 0 and, as`x to oo, y to -oo#.

graph{(1/sqrt(x^2+e^x)-xe^x-y)(y-1.364)(x+.555+.001y)=0 [-10, 10, -5, 5]}