X^3-2x-2cos(x) find local extrema

In summary, the problem is to find the local extrema of the function f(x)=(x^3-2x-2cos(x)). This can be done by setting the first derivative equal to zero and solving for x. However, the resulting equation cannot be solved algebraically and must be approached graphically.
  • #1
physicsdreams
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Homework Statement

Find the local extrema of the function f(x)=(x^3-2x-2cos(x))

Homework Equations



derivatives, some algebra

The Attempt at a Solution



Well, the concept is simple.
Solve for the first derivative and set it equal to zero:

dy/dx=2sin(x)+3x^2-2=0 and

Next, solve for x to determine the "critical points".

My problem is in solving this seemingly simple equation algebraically.
I can simplify it to:
sin(x)=1-(3x^2)/2 (which doesn't help).

I have a feeling that it's not possible to solve algebraically, (but that I can still graph it).
Can anyone confirm my suspicion?

Thanks in advance!
 
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  • #2
No, you can't solve that algebraically. Proceed with a graphical solution.
 
  • #3
Thanks!
 

1. What is the definition of "local extrema"?

Local extrema are points on a graph where the function reaches its highest or lowest value in a specific region, but not necessarily the highest or lowest value on the entire graph.

2. How do you find the local extrema of a function?

To find the local extrema of a function, you must first take the derivative of the function and set it equal to zero. Then, solve for the critical points by finding the roots of the derivative. Finally, plug these critical points into the original function to determine if they are local maxima or minima.

3. What is the significance of finding local extrema in a function?

Finding local extrema can help us identify where a function is increasing or decreasing, as well as the maximum and minimum values of the function within a specific region. This information is useful in various real-world applications, such as optimization problems.

4. Can a function have more than one local extremum?

Yes, a function can have multiple local extrema. For example, a quadratic function can have both a local maximum and minimum.

5. How does the presence of trigonometric functions, such as cos(x), affect the local extrema of a function?

The presence of trigonometric functions can create more complex local extrema, as they introduce periodic behavior in the function. This means that the local extrema may occur at multiple points within a specific region, rather than just one point. Additionally, the amplitude and period of the trigonometric function can impact the height and spacing of the local extrema.

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