The function is continuous for all real numbers

MATHEMATICS · Single correct · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

The function

f(x)=\sqrt[3]{x}

is continuous for all real numbers. Which statement best describes the behavior of

f'

x=0

? (a)

f'(0)=0

because the graph is flat at the origin. (b)

f'(0)=\infty

since

f'(x)=\dfrac13x^{-2/3}

and substituting

x=0

gives division by

0

. (c)

f'

does not exist at

x=0

because the tangent line is vertical. (d)

f'

does not exist at

x=0

because

f

is not continuous there.