Question

In Exercises 40 and \(41,\) let \(f(x)=\left\\{\begin{array}{ll}{x^{2}-1,} & {x<0} \\ {2 x-1,} & {x \geq 0}\end{array}\right.\) Multiple Choice Which of the following is equal to the left-hand derivative of \(f\) at \(x=0 ?\) \((\mathbf{A})-2 \quad(\mathbf{B}) 0 \quad(\mathbf{C}) 2 \quad(\mathbf{D}) \propto(\mathbf{E})-\infty\)

Short answer

The left-hand derivative of \(f(x)\) at \(x=0\) is 0, corresponding to option B.

Step-by-step solution

Recognize the relevant section of the piecewise function

As the exercise is asking for the left-hand derivative at \(x = 0\), the relevant part of the function is \(x^{2}-1\) since \(x<0\) here.

Calculate the derivative

The derivative of the function \(x^{2}-1\) is \(2x\). When we evaluate the derivative at \(x = 0\), we obtain \(2(0) = 0\)

Match the derivative with the options

Having calculated the left-hand derivative as 0, you can see that it corresponds to option B.