Question

If f(x)=1+x23, which of the following is NOT true?
a. f is continuous for all real numbers.
b. f has a minimum at x=0.
c. f is increasing for x>0.
d. f'(x) exists for all x
e. f''(x) is negative for x>0.

Answer 1

The radical function $f(x) = 1 + x^{2/3}$ has the key characteristic of being undefined at $x = 0$. $f'(x)$ does not exists for $x = 0$.  (Choice: D)

Let be $f(x) = 1 + x^{2/3}$, which represents a radical function, whose characteristics are described below:

1) All odd radical functions are continuous for all numbers, this is, $f(x)$ exists for every $x \in\mathbb{R}$.

2) $f(x)$ has an absolute minimum at $x = 0$.

3) $f(x)$ increases for $x> 0$ or $x < 0$.

4) First derivative does not exists for $x = 0$.

5) The first and second derivatives of the function are $f'(x) = \frac{2}{3}\cdot x^{-\frac{1}{3} }$ and $f''(x) = -\frac{2}{9}\cdot x^{-\frac{4}{3} }$, which means that $f''(x) < 0$ for $x > 0$.

Hence, we infer that $f'(x)$ does not exists for $x = 0$.

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