Which of the following describes the behavior of y=(x+2)^(1/3)

a) cusp

b) corner

c) vertical tangent

how do you prove it/figure it out?

- Dec 15th 2009, 09:10 PMcrazWhich of the following describes the behavior of y=(x+2)^(1/3(
Which of the following describes the behavior of y=(x+2)^(1/3)

a) cusp

b) corner

c) vertical tangent

how do you prove it/figure it out? - Dec 15th 2009, 09:35 PMoblixps
if you graph it out, its a vertical tangent.

if you want to find what it is algebraically though, take the derivative and see where its undefined. the derivative is undefined at x = -2.

now take the right and left hand limits of the derivative of (x+2)6(1/3) as x approaches -2. if you plug in -2 directly you are going to get 1/0 which tells you that it is +/- infinite. if you plug in numbers very close to -2, since the (x+2) is being squared (and then take the cube root), the denominator will always be positive so both limits approach +infinite. since this is the case, it must be a vertical tangent. graph the function out and see if this makes sense geometrically. - Dec 16th 2009, 02:49 AMmr fantastic
I assume you want to know the natue of the point (-2, 0). It's the same as the nature of the point (0, 0) on the curve $\displaystyle y = x^{1/3}$. With reference to that curve, note that:

1. The domain is all real numbers.

2. The curve is monotonically increasing.

3. The derivative is undefined at x = 0.