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?
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?
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.
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.