Damion Ellis

2023-02-18

What are the absolute extrema of $f\left(x\right)={x}^{3}-3x+1\in [0,3]$?

Cheyenne Lynn

Beginner2023-02-19Added 10 answers

There are two potential candidates for an interval's absolute extrema. They are the endpoints of the interval (here, $0$ and $3$) and the critical values of the function located within the interval.

The critical values can be found by finding the function's derivative and finding for which values of $x$ it equals $0$.

We can use the power rule to find that the derivative of $f\left(x\right)={x}^{3}-3x+1$ is $f\prime \left(x\right)=3{x}^{2}-3$.

The critical values are when $3{x}^{2}-3=0$, which simplifies to be $x=\pm 1$. However, $x=-1$ is not in the interval so the only valid critical value here is the one at $x=1$. We now know that the absolute extrema could occur at $x=0,x=1,$ and $x=3$.

To determine which is which, plug them all into the original function.

$f\left(0\right)=1$

$f\left(1\right)=-1$

$f\left(3\right)=19$

From here we can see that there is an absolute minimum of $-1$ at $x=1$ and an absolute maximum of $19$ at $x=3$.

Check the function's graph:

graph{x^3-3x+1 [-0.1, 3.1, -5, 20]}

The critical values can be found by finding the function's derivative and finding for which values of $x$ it equals $0$.

We can use the power rule to find that the derivative of $f\left(x\right)={x}^{3}-3x+1$ is $f\prime \left(x\right)=3{x}^{2}-3$.

The critical values are when $3{x}^{2}-3=0$, which simplifies to be $x=\pm 1$. However, $x=-1$ is not in the interval so the only valid critical value here is the one at $x=1$. We now know that the absolute extrema could occur at $x=0,x=1,$ and $x=3$.

To determine which is which, plug them all into the original function.

$f\left(0\right)=1$

$f\left(1\right)=-1$

$f\left(3\right)=19$

From here we can see that there is an absolute minimum of $-1$ at $x=1$ and an absolute maximum of $19$ at $x=3$.

Check the function's graph:

graph{x^3-3x+1 [-0.1, 3.1, -5, 20]}

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function

$f(x,y)={x}^{3}-6xy+8{y}^{3}$ $\frac{1}{\mathrm{sec}(x)}$ in derivative?

What is the derivative of $\mathrm{ln}(x+1)$?

What is the limit of $e}^{-x$ as $x\to \infty$?

What is the derivative of $f\left(x\right)={5}^{\mathrm{ln}x}$?

What is the derivative of $e}^{-2x$?

How to find $lim\frac{{e}^{t}-1}{t}$ as $t\to 0$ using l'Hospital's Rule?

What is the integral of $\sqrt{9-{x}^{2}}$?

What is the derivative of $f\left(x\right)=\mathrm{ln}\left[{x}^{9}{(x+3)}^{6}{({x}^{2}+7)}^{5}\right]$ ?

What Is the common difference or common ratio of the sequence 2, 5, 8, 11...?

How to find the derivative of $y={e}^{5x}$?

How to evaluate the limit $\frac{\mathrm{sin}\left(5x\right)}{x}$ as x approaches 0?

How to find derivatives of parametric functions?

What is the antiderivative of $-5{e}^{x-1}$?

How to evaluate: indefinite integral $\frac{1+x}{1+{x}^{2}}dx$?