Finding Local Extrema Using Second-Derivative Test

Hi Ayliz, let's solve this together. We want to find the local maxima and minima of the function f of x equals x cubed minus six x squared plus nine x plus one using the second-derivative test.

To do this, we will first find the critical numbers by setting the first derivative to zero, and then we will apply the second-derivative test to classify them.

Let's first take the derivative of our function with respect to x. Using the power rule on each term, we write down the derivative expression.

Differentiating term by term, we get f prime of x equals three x squared minus twelve x plus nine.

Now, we set f prime of x equal to zero to find the critical numbers.

We can factor out a three from the entire equation.

Next, we divide both sides by three to simplify the quadratic.

This quadratic factors nicely into x minus one times x minus three equals zero.

Thus, our critical numbers are x equals one and x equals three.

Now let's find the second derivative, f double prime of x, by taking the derivative of f prime of x.

Differentiating, we obtain f double prime of x equals six x minus twelve.

Now we test our first critical number, x equals one, by substituting it into the second derivative.