Question

The derivative of $$f(x)=|x^3|$$ at x = 0 is
1) 0
2) 1
3) -1
4) not defined

Answer 1

Hint: Here, we are given a function and we need to find $$f^{\prime }(x)$$ at x = 0. First, we will find the value of the given function when the mode sign is removed. Then we will find the derivative from the given formula $$f^{\prime }(x)=\underset{h\to 0}{lim}{\displaystyle \frac{f(x+h)-f(x)}{h}}$$ . Using this formula, we will find the values of left hand limit (LHL) and right hand limit (RHL). If both LHL = RHL then the limit of the function exists and we will get the final output.

Complete step-by-step answer:
Given that,
$$f(x)=|x^3|$$
Thus, to open the mode sign, we will have the following values of x as below,
$$f(x)=-x^3,x<0$$ and $$f(x)=x^3,x⩾0$$
Since our given function is continuous, because
Left hand limit (LHL) = Right hand limit (RHL) at x = 0.
$$f^{\prime }(x)=3x^2$$ at $$x⩾0$$ and $$f^{\prime }(x)=-3x^2$$ at x<0
We will find the derivative through the given formula,
$$f^{\prime }(x)=\underset{h\to 0}{lim}{\displaystyle \frac{f(x+h)-f(x)}{h}}$$
First, for $$x⩾0$$, then we will have,
$$\therefore f^{\prime }(0)=\underset{h\to 0^{+}}{lim}{\displaystyle \frac{f(0+h)-f(0)}{h}}$$
$$=\underset{h\to 0^{+}}{lim}{\displaystyle \frac{h^3-0}{h}}$$
$$=\underset{h\to 0^{+}}{lim}{\displaystyle \frac{h^3}{h}}$$
$$=\underset{h\to 0^{+}}{lim}{h}^2$$
$$=0$$
Next, for x<0, then we will have,
$$\therefore f^{\prime }(0)=\underset{h\to 0^{-}}{lim}{\displaystyle \frac{f(0-h)-f(0)}{h}}$$
$$=\underset{h\to 0^{+}}{lim}{\displaystyle \frac{-h^3-0}{h}}$$
$$=\underset{h\to 0^{+}}{lim}{\displaystyle \frac{-h^3}{h}}$$
$$=\underset{h\to 0^{+}}{lim}(-h^2)$$
$$=0$$
Since, LHL = RHL. This means that the limit exists.
Thus the function is derivable.
Hence, for the given function $$f(x)=|x^3|$$ at x = 0, $$f^{\prime }(0)=0$$.
So, the correct answer is “Option B”.

Note: A limit is defined as a value that a function approaches the output for the given input values. It is used in the analysis process, and it always concerns the behaviour of the function at a particular point. A derivative is defined as the instantaneous rate of change in function based on one of its variables. It is similar to finding the slope of a tangent to the function at a point. Integration is a method to find definite and indefinite integrals.