1. ## questions about cusps and tangent lines in Derivatives

these questions i have answeared uncorrectly and don't understand why i'm wrong, anyway here are the questions:

Determine whether f has :
(a)a vertical tangent line at (0,0)
(b)a cusp at (0,0)

(1)f(x) = x ^ (1/3)
Correct Answear: (a)yes (b) no
My Answear: (a)no (b)no

(2)f(x) = x ^ (2/5)
Correct Answear: (a)yes (b)yes
My Answear: (a)no (b)no

(3)f(x) = 5x ^ (3/2)
Correct Answear: (a)no (b)no
My Answear: (a)yes (b)no

if there is a cusp at a point then the right hand limit = infinity and the left hand limit = infinity with opposite signs at that point ?
and a vrtical tangent line means the dericative of f at this point = infinity?

Determine whether f has:
(a) a vertical tangent line at (0,0)
(b) a cusp at (0,0)

(1) f(x) = x^{1/3}

Correct answer: (a) yes (b) no
My answer: (a) no (b) no

f'(x) .= .(1/3)x^{-2/3} .= .(1/3)[x^{-1/3}]^2

(a) f'(0) is undefined; there is a vertical tangent.

(b) The right-hand limit and the left-hand limit are both +∞
. . .There is no cusp there.

(2) f(x) = x^{2/5}
Correct answer: (a) yes (b) yes
My answer: (a) no (b) no

f'(x) .= .(2/5)x^{-3/5}

(a) f(0) is undefined; there is a vertical tangent.

(b) The left-hand limit is -∞ . . . The right-hand limit is +∞
. . .There is a cusp.

(3) f(x) = 5x^{3/2}

Correct answer: (a) no (b) no
My answer: (a) yes (b) no

f'(x) .= .(15/2)x^{1/2}

(a) f'(0) = 0 is defined . . . no vertical tangent.

(b) There is no left-hand limit . . . no cusp.
. . .(x cannot approach 0 from the left.)

3. thanks Soroban for the reply, i have graphed those three functions and have some questions about ur reply.about the first answear isn't both the left hand limit and right hand limit for f prime is undefined? i mean how could u tell if it is +∞ or -∞?
as for the second answear (The left-hand limit is -∞ . . . The right-hand limit is +∞
...There is a cusp.
)both graphs of the derivative for the first and second questions look similiar how could u define one limit as +∞ and the second limit as -∞ (that's for the second question)??