Question

To answer each question, refer to the following basic graphs. Which one is the graph of ƒ(x)=∛x? Is there any open interval over which the function is decreasing?

Accepted Answer

Recall that the function $$ f(x) = \sqrt[3]{x}  is $$the cube root function, which is defined for all real numbers $$ x . $$Its graph passes through the origin $$ (0,0) $$ and has a characteristic S-shape, increasing from left to right. Identify the graph of $$ f(x) = \sqrt[3]{x}  by $$looking for the curve that is continuous for all $$ x $$, passes through the origin, and increases slowly for negative $$ x $$ values (going down to the left) and increases for positive $$ x $$ values (going up to the right). To determine if the function is decreasing on any open interval, consider the derivative of $$ f(x) = \sqrt[3]{x} . $$The derivative is $$ f'(x) = \frac{1}{3x^{2/3}} $$, which is positive for all $$ x 
eq 0 . $$Since $$ f'(x) \u003e 0 $$ for all $$ x 
eq 0 $$, the function is strictly increasing everywhere except possibly at $$ x = 0 $$, where the derivative is undefined but the function is still increasing through that point. Therefore, there is no open interval over which $$ f(x) = \sqrt[3]{x}  is $$decreasing; it is increasing on its entire domain.

To answer each question, refer to the following basic graphs. Which one is the graph of ƒ(x)=∛x? Is there any open interval over which the function is decreasing?

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Cube Root Function

Graph Interpretation

Increasing and Decreasing Intervals