Hello, mHadad!

Your understanding is correct, but your answers were incorrect.

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; thereisa 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; thereisa vertical tangent.

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

. . .Thereisa 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 . . .novertical tangent.

(b) There is no left-hand limit . . . no cusp.

. . .(xcannot approach 0 from the left.)