Question

Writing to Learn Let \(f(x)=\left|x^{3}-9 x\right|\) (a) Does \(f^{\prime}(0)\) exist? (b) Does \(f^{\prime}(3)\) exist? (c) Does \(f^{\prime}(-3)\) exist? (d) Determine all extrema of \(f\)

Short answer

\(f^{\prime}(0)\) does not exist, \(f^{\prime}(3)\) does not exist. \(f'(-3)\) does exists. There is a maximum at \(x=3\), a minimum at \(x=-3\), and a saddle point at \(x=0\)

Step-by-step solution

Identify the function inside modulus

First thing is to look at the function \(f(x)=\|x^{3}-9x\|\). The function inside modulus is \(x^{3}-9x\). We will analyze when this function changes sign that will effectively change the behavior of the modulus function.

Find out when the function changes sign

Setting \(x^{3}-9x\) to 0 will give us the points where the function changes the sign. The equation will become \(x(x^{2}-9)=0\), from which we get \(x=-3, 0, 3\). These are the points where the sign changes and the modulus function splits.

Calculate the derivative

For \(x<-3\) we get \(f'(x)=-(3x^2-9)\), for \(-33\) we get \(f'(x)=3x^2-9\). You can easily calculate these by noticing the negative sign for the modulus function.

Answer part (a)

By substitute 0 in \(f'(x)\) which will get \(f'(0)= -9 \) and \( f'(0)= 9\), so \(f^{\prime}(0)\) does not exist because the left hand limit does not equal to the right hand limit.

Answer part (b)

For this part, substitute 3 in \(f'(x)\) which will get \(f'(3)= 0 \) and \( f'(3)= 18\), so \(f^{\prime}(3)\) does not exist since the left hand limit does not equal to the right hand limit.

Answer part(c)

By substitute in -3 in the derivative, \(f'(-3)=0 = f'(-3)\) so \(f'(-3)\) does exists, the left hand limit equals to the right hand limit

Answer part(d)

To find the extremas you need to find the points where the derivative is 0 or undefined. We already know \(f'(0)\) and \(f'(3)\) does not exist, so \(x=0,3\) are candidates for extrema. To check if its maximum/minimum or a saddle point we will also need to calculate \(f''(x)\). Also \(f'(-3)\) equals 0. By further analyszing \(f''(x)\) it will be realize that there is a maximum at \(x=3\), a minimum at \(x=-3\), and a saddle point at \(x=0\)