Question

How do you find intercepts, extrema, points of inflections, asymptotes and graph `f(x)=(4x)/(sqrt(x^2+15))`?

Answer 1

No intercepts. Horizontal asymptotes : `rarr y = 4 and y =-4 larr`
Point of inflexion : O ( see the graph ),
No extrema, as `y in (-oo, oo)`.

Explanation:

Use `y = f(x) = ( 4 x )/(|x| sqrt(1+15/x^2)`
If (x, y) is on the graph, so is (-x, -y), showing symmetry about O.

As `x to +-oo, y to +-4`

Cross multiply and differentiate to get the following results.

y' simplifies to `60/(x^2+15)^1.5 in (0, 4sqrt15]`

y, y and y''. vanish at O and `y''' ne 0`, revealing O as the POI..

graph{(4x/sqrt(x^2+15)-y)(y+4)(y-4)=0 [-10, 10, -5, 5]}