Question

Which of the extrema below does the the function f(x) = x5 - 2x³ + 1 have?I. absolute maximumII. absolute minimumIII. local maximumIV. local minimumA. III and IV onlyB. I and II onlyOC. I and III onlyD. I, II, III, and IV

Answer 1

First, we need to calculate the critical points of the function. The critical points are the zeros of the first derivative. The first derivative of our function can be found by using the power rule:

`f^(\prime)(x)=5x^4-6x^2`

The critical points of our function are the solutions for the following equation:

`5x^4-6x^2=0`

x = 0 is a solution. If we ignore this solution, we can find the other solutions by dividing all the terms in our equation by x².

`\begin{gathered} 5x^2-6=0 \\ x=\pm\sqrt{(6)/(5)} \end{gathered}`

We have 3 critical points in our function(zero has multiplicity 2).

The maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the local or relative extrema), or on the entire domain (the global).

To known if our critical points are local or global we need to analyze the end behavior of our function.

`\begin{gathered} \lim_(x\to\infty)f(x)=\infty \\ \lim_(x\to-\infty)f(x)=-\infty \end{gathered}`

Since the function diverges on the negative end to minus infinity and diverges on the positive end to infinity, this functions has no global minimum or maximum.

To find if our values are maximum or minimum we evaluate them in our function and check if the interval between the is increasing or decreasing.

`\begin{gathered} x=-\sqrt{(6)/(5)}\to local\:maximum \\ x=0\to inflection\:point \\ x=\sqrt{(6)/(5)}\to local\:minimum \end{gathered}`

Then, the answer for our problem is option A. We have only local maximum and minimum.