Calculus Examples

Find the Absolute Max and Min over the Interval f(x)=1/x , x>=1
,
Step 1
Find the critical points.
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Step 1.1
Find the first derivative.
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Step 1.1.1
Find the first derivative.
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Step 1.1.1.1
Rewrite as .
Step 1.1.1.2
Differentiate using the Power Rule which states that is where .
Step 1.1.1.3
Rewrite the expression using the negative exponent rule .
Step 1.1.2
The first derivative of with respect to is .
Step 1.2
Set the first derivative equal to then solve the equation .
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Step 1.2.1
Set the first derivative equal to .
Step 1.2.2
Set the numerator equal to zero.
Step 1.2.3
Since , there are no solutions.
No solution
No solution
Step 1.3
Find the values where the derivative is undefined.
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Step 1.3.1
Set the denominator in equal to to find where the expression is undefined.
Step 1.3.2
Solve for .
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Step 1.3.2.1
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 1.3.2.2
Simplify .
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Step 1.3.2.2.1
Rewrite as .
Step 1.3.2.2.2
Pull terms out from under the radical, assuming positive real numbers.
Step 1.3.2.2.3
Plus or minus is .
Step 1.4
Evaluate at each value where the derivative is or undefined.
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Step 1.4.1
Evaluate at .
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Step 1.4.1.1
Substitute for .
Step 1.4.1.2
The expression contains a division by . The expression is undefined.
Undefined
Undefined
Undefined
Step 1.5
There are no values of in the domain of the original problem where the derivative is or undefined.
No critical points found
No critical points found
Step 2
Evaluate at the included endpoints.
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Step 2.1
Evaluate at .
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Step 2.1.1
Substitute for .
Step 2.1.2
Divide by .
Step 2.2
List all of the points.
Step 3
Since there is no value of that makes the first derivative equal to , there are no local extrema.
No Local Extrema
Step 4
Compare the values found for each value of in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest value and the minimum will occur at the lowest value.
Absolute Maximum:
No absolute minimum
Step 5