Minimize the function 4*x + 1/x
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Find the positive number which minimizes
In general, how do we find a local minimum of a function?
We find the derivative of the function and set it to 0. The corresponding value of x minimizes the function.
Set up the derivative.
How do we handle the derivative of the reciprocal?
We convert it to the negative first power.
Set up the derivative that way.
Do the differentiation.
Set this to 0 and solve.
We get
How do we take care of the negative power?
We take the reciprocal of each side of the equation.
Do that.
We have
Solve for x
What is the interpretation of these values?
They are the values for which the function is an extremum.
We were asked for the positive number at which this function has a local minimum. Which value should we use?
How can we check that this is a local minimum, not a local maximum.
We can find the second derivative of the original function at
,
and check the algebraic sign.
Do that.
Evaluate this at
Does this indicate a local maximum or a local minimum?
Since the result is positive, indicating that the slope is increasing, we have a local minimum at
.
Plot the function f(x) for –2 < x < 2. Then compare it with the graph below.
The graph of the function, showing the local minimum at x = 1/2
Does the curve show a minimum where we expected it?
Yes, at x = 0.5.
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