Question

Find the derivative of \(f(x)=x|x| .\) Does \(f^{\prime \prime}(0)\) exist?

Short answer

The derivative of the given function is \(f'(x)=2x\) when \(x \geq 0\) and \(f'(x)=-2x\) when \(x < 0\). The second derivative, \(f''(x)=2\) for all \(x\), exists at \(x=0\) and is equal to 2.

Step-by-step solution

Identify the Function

The function given here is \(f(x)=x|x|\), which is a composite function of the absolute value function and a linear function.

Calculate the First Derivative

To compute the first derivative of this function, we should notice that the function can be written as: \(x|x| = x^2\) when \(x \geq 0\), and \(x|x| = -x^2\) when \(x < 0\). The derivative is thus: \(f'(x)=2x\) when \(x \geq 0\) and \(f'(x)=-2x\) when \(x < 0\). At \(x=0\), \(f'(x)\) is 0.

Compute the Second Derivative

The second derivatives for each part would be \(f''(x)=2\) when \(x \geq 0\) and \(f''(x)=2\) when \(x < 0\). This would imply that the second derivative at \(x=0\) does exist and is 2.