Finding Values given only local max and min.

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In summary, the conversation discusses finding the values of a and b for a function with a local maximum at x = -4 and a local minimum at x = 5. The conversation also touches on the concepts of extrema, derivatives, and the relationship between derivatives and increasing or decreasing functions. The solution involves setting the derivative of the function equal to 0 at the given points and solving for a and b.
  • #1
Hypnos_16
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Homework Statement


Find the values of a and b if the function f(x) = 2x3 + ax2 + bx + 36 has a local maximum when x = −4 and a local minimum when x = 5.


Homework Equations



I'm not even sure how to start this, it's just baffling me for some reason

The Attempt at a Solution



i do not have one... sorry.
 
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  • #2
extremum occur at points where the derivative is zero (why?)
 
  • #3
Because those are the points at which the function's sigh changes?
 
  • #4
No. That's a zero of the function itself, not of the derivative. What is an extremum, and how looks the function's derivative at an extremum?
 
  • #5
Okay you've completely lost me there.
 
  • #6
I think when they gave you this problem they have explained maxima and minima (together called extrema) before in class, haven't they? If not, you really should look it up in any book on calculus (the explanations on Wikipedia are really not that good).

To continue, try to find the answer to the following questions (no need for formulas, just plain english):
a) What is a maximum of a function?
b) What does the derivative of a function mean?
c) If a function is increasing, what does that tell us about its derivative?
d) And if it is decreasing?
e) And at a maximum?
 
  • #7
a). Maximum is the point on the graph where the function is the highest and the graph is continuous
b). Change in the Function with respect to one of the variables
c). The derivative is also increasing
d). The derivative is also decreasing
e). The derivative is at it's highest possible value
 
  • #8
a) and b) are true, but the rest are not.
If a function is increasing, then just as you said the derivative gives the change in the function. So if the function is increasing the derivative is positive, and if the function is decreasing the derivative is negative.
Now imagine a maximum of f at the point x_0. Just left of it, the function is increasing, and just right of it, the function is decreasing. So for x < x_0, the derivative is positive, and for x > x_0, the derivative is negative. If the derivative is continous, what must be its value at x_0?
 
  • #9
the value of x_0 must be 0 then.
 
  • #10
There is a local minimum at [tex]f'(x)=0[/tex]
 
  • #11
Right, the derivative must be 0 zero at a maximum. Same goes for a minimum. So if your original question said f(x) has a maximum when x=-4, you calculate the derivative f'(x) and then set f'(x=-4) = 0. Same goes for the minimum at x= 5. Then you have two equations with two unknowns a and b and can solve for them. If everything is clearer now, you can do the calculations and post the equations and the results so we can check them.
 

1. What is the purpose of finding values given only local max and min?

The purpose of finding values given only local max and min is to determine the behavior and characteristics of a function at specific points. This information can be used to analyze data, make predictions, and understand the overall nature of the function.

2. How do I identify local max and min points on a graph?

Local maximum points are the highest points on the graph within a particular interval, while local minimum points are the lowest points on the graph within a specific interval. These points can be identified by looking for where the graph changes from increasing to decreasing (local max) or from decreasing to increasing (local min).

3. Can a function have more than one local max or min?

Yes, a function can have multiple local max and min points. This occurs when the graph has multiple peaks and valleys within a given interval. Each local max is the highest point and each local min is the lowest point within its respective interval.

4. How can I use the values of local max and min to determine the overall shape of the graph?

The values of local max and min can be used to determine the overall shape of the graph by connecting them with a smooth curve. This curve will show the general trend of the function and can help identify any patterns or symmetries present.

5. Are local max and min points the only important points on a graph?

No, local max and min points are not the only important points on a graph. Other important points include the global max and min (the highest and lowest points on the entire graph), points of inflection (where the graph changes from concave up to concave down or vice versa), and intercepts (where the graph crosses the x or y-axis).

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