Finding Values given only local max and min.

[Hypnos_16]

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.

 

[lanedance]

extremum occur at points where the derivative is zero (why?)

 

[Hypnos_16]

Because those are the points at which the function's sigh changes?

 

[grey_earl]

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?

 

[Hypnos_16]

Okay you've completely lost me there.

 

[grey_earl]

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?

 

[Hypnos_16]

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

 

[grey_earl]

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?

 

[Hypnos_16]

the value of x_0 must be 0 then.

 

[hunt_mat]

There is a local minimum at $$f'(x)=0$$

 

[grey_earl]

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.