Algebra Examples

Find the Maximum/Minimum Value y=-1+6cos((2pi)/7(x-5))
Step 1
Find the first derivative of the function.
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Step 1.1
Differentiate.
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Step 1.1.1
By the Sum Rule, the derivative of with respect to is .
Step 1.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 1.2
Evaluate .
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Step 1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.2
Differentiate using the chain rule, which states that is where and .
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Step 1.2.2.1
To apply the Chain Rule, set as .
Step 1.2.2.2
The derivative of with respect to is .
Step 1.2.3
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.4
By the Sum Rule, the derivative of with respect to is .
Step 1.2.5
Differentiate using the Power Rule which states that is where .
Step 1.2.6
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.7
Add and .
Step 1.2.8
Multiply by .
Step 1.2.9
Combine and .
Step 1.2.10
Multiply by .
Step 1.2.11
Combine and .
Step 1.2.12
Multiply by .
Step 1.2.13
Move the negative in front of the fraction.
Step 1.3
Simplify.
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Step 1.3.1
Apply the distributive property.
Step 1.3.2
Combine terms.
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Step 1.3.2.1
Combine and .
Step 1.3.2.2
Multiply by .
Step 1.3.2.3
Move the negative in front of the fraction.
Step 1.3.2.4
Combine and .
Step 1.3.2.5
Subtract from .
Step 2
Find the second derivative of the function.
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Step 2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.2
Differentiate using the chain rule, which states that is where and .
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Step 2.2.1
To apply the Chain Rule, set as .
Step 2.2.2
The derivative of with respect to is .
Step 2.2.3
Replace all occurrences of with .
Step 2.3
Differentiate.
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Step 2.3.1
Combine and .
Step 2.3.2
Move to the left of .
Step 2.3.3
By the Sum Rule, the derivative of with respect to is .
Step 2.3.4
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.5
Differentiate using the Power Rule which states that is where .
Step 2.3.6
Multiply by .
Step 2.3.7
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.8
Combine fractions.
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Step 2.3.8.1
Add and .
Step 2.3.8.2
Multiply by .
Step 2.3.8.3
Multiply by .
Step 2.4
Raise to the power of .
Step 2.5
Raise to the power of .
Step 2.6
Use the power rule to combine exponents.
Step 2.7
Add and .
Step 2.8
Multiply by .
Step 3
To find the local maximum and minimum values of the function, set the derivative equal to and solve.
Step 4
Set the numerator equal to zero.
Step 5
Solve the equation for .
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Step 5.1
Divide each term in by and simplify.
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Step 5.1.1
Divide each term in by .
Step 5.1.2
Simplify the left side.
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Step 5.1.2.1
Cancel the common factor of .
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Step 5.1.2.1.1
Cancel the common factor.
Step 5.1.2.1.2
Rewrite the expression.
Step 5.1.2.2
Cancel the common factor of .
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Step 5.1.2.2.1
Cancel the common factor.
Step 5.1.2.2.2
Divide by .
Step 5.1.3
Simplify the right side.
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Step 5.1.3.1
Cancel the common factor of and .
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Step 5.1.3.1.1
Factor out of .
Step 5.1.3.1.2
Cancel the common factors.
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Step 5.1.3.1.2.1
Factor out of .
Step 5.1.3.1.2.2
Cancel the common factor.
Step 5.1.3.1.2.3
Rewrite the expression.
Step 5.1.3.2
Divide by .
Step 5.2
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 5.3
Simplify the right side.
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Step 5.3.1
The exact value of is .
Step 5.4
Add to both sides of the equation.
Step 5.5
Since the expression on each side of the equation has the same denominator, the numerators must be equal.
Step 5.6
Divide each term in by and simplify.
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Step 5.6.1
Divide each term in by .
Step 5.6.2
Simplify the left side.
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Step 5.6.2.1
Cancel the common factor of .
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Step 5.6.2.1.1
Cancel the common factor.
Step 5.6.2.1.2
Rewrite the expression.
Step 5.6.2.2
Cancel the common factor of .
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Step 5.6.2.2.1
Cancel the common factor.
Step 5.6.2.2.2
Divide by .
Step 5.6.3
Simplify the right side.
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Step 5.6.3.1
Cancel the common factor of and .
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Step 5.6.3.1.1
Factor out of .
Step 5.6.3.1.2
Cancel the common factors.
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Step 5.6.3.1.2.1
Factor out of .
Step 5.6.3.1.2.2
Cancel the common factor.
Step 5.6.3.1.2.3
Rewrite the expression.
Step 5.6.3.2
Cancel the common factor of .
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Step 5.6.3.2.1
Cancel the common factor.
Step 5.6.3.2.2
Divide by .
Step 5.7
The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from to find the solution in the second quadrant.
Step 5.8
Solve for .
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Step 5.8.1
Subtract from .
Step 5.8.2
Move all terms not containing to the right side of the equation.
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Step 5.8.2.1
Add to both sides of the equation.
Step 5.8.2.2
To write as a fraction with a common denominator, multiply by .
Step 5.8.2.3
Combine and .
Step 5.8.2.4
Combine the numerators over the common denominator.
Step 5.8.2.5
Simplify the numerator.
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Step 5.8.2.5.1
Move to the left of .
Step 5.8.2.5.2
Add and .
Step 5.8.3
Since the expression on each side of the equation has the same denominator, the numerators must be equal.
Step 5.8.4
Divide each term in by and simplify.
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Step 5.8.4.1
Divide each term in by .
Step 5.8.4.2
Simplify the left side.
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Step 5.8.4.2.1
Cancel the common factor of .
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Step 5.8.4.2.1.1
Cancel the common factor.
Step 5.8.4.2.1.2
Rewrite the expression.
Step 5.8.4.2.2
Cancel the common factor of .
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Step 5.8.4.2.2.1
Cancel the common factor.
Step 5.8.4.2.2.2
Divide by .
Step 5.8.4.3
Simplify the right side.
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Step 5.8.4.3.1
Cancel the common factor of .
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Step 5.8.4.3.1.1
Cancel the common factor.
Step 5.8.4.3.1.2
Rewrite the expression.
Step 5.9
The solution to the equation .
Step 6
Evaluate the second derivative at . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.
Step 7
Evaluate the second derivative.
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Step 7.1
Multiply by .
Step 7.2
Simplify the numerator.
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Step 7.2.1
Combine the numerators over the common denominator.
Step 7.2.2
Subtract from .
Step 7.2.3
Divide by .
Step 7.2.4
The exact value of is .
Step 7.2.5
Multiply by .
Step 8
is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.
is a local maximum
Step 9
Find the y-value when .
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Step 9.1
Replace the variable with in the expression.
Step 9.2
Simplify the result.
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Step 9.2.1
Simplify each term.
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Step 9.2.1.1
Subtract from .
Step 9.2.1.2
Multiply by .
Step 9.2.1.3
The exact value of is .
Step 9.2.1.4
Multiply by .
Step 9.2.2
Add and .
Step 9.2.3
The final answer is .
Step 10
Evaluate the second derivative at . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.
Step 11
Evaluate the second derivative.
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Step 11.1
Simplify the numerator.
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Step 11.1.1
Combine and .
Step 11.1.2
Combine and .
Step 11.2
Multiply by .
Step 11.3
Reduce the expression by cancelling the common factors.
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Step 11.3.1
Reduce the expression by cancelling the common factors.
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Step 11.3.1.1
Factor out of .
Step 11.3.1.2
Factor out of .
Step 11.3.1.3
Cancel the common factor.
Step 11.3.1.4
Rewrite the expression.
Step 11.3.2
Divide by .
Step 11.4
Simplify the numerator.
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Step 11.4.1
Combine the numerators over the common denominator.
Step 11.4.2
Subtract from .
Step 11.4.3
Cancel the common factor of .
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Step 11.4.3.1
Cancel the common factor.
Step 11.4.3.2
Divide by .
Step 11.4.4
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.
Step 11.4.5
The exact value of is .
Step 11.4.6
Multiply by .
Step 11.4.7
Multiply by .
Step 11.5
Move the negative in front of the fraction.
Step 11.6
Multiply .
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Step 11.6.1
Multiply by .
Step 11.6.2
Multiply by .
Step 12
is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.
is a local minimum
Step 13
Find the y-value when .
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Step 13.1
Replace the variable with in the expression.
Step 13.2
Simplify the result.
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Step 13.2.1
Simplify each term.
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Step 13.2.1.1
To write as a fraction with a common denominator, multiply by .
Step 13.2.1.2
Combine and .
Step 13.2.1.3
Combine the numerators over the common denominator.
Step 13.2.1.4
Simplify the numerator.
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Step 13.2.1.4.1
Multiply by .
Step 13.2.1.4.2
Subtract from .
Step 13.2.1.5
Cancel the common factor of .
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Step 13.2.1.5.1
Factor out of .
Step 13.2.1.5.2
Cancel the common factor.
Step 13.2.1.5.3
Rewrite the expression.
Step 13.2.1.6
Cancel the common factor of .
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Step 13.2.1.6.1
Cancel the common factor.
Step 13.2.1.6.2
Rewrite the expression.
Step 13.2.1.7
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.
Step 13.2.1.8
The exact value of is .
Step 13.2.1.9
Multiply .
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Step 13.2.1.9.1
Multiply by .
Step 13.2.1.9.2
Multiply by .
Step 13.2.2
Subtract from .
Step 13.2.3
The final answer is .
Step 14
These are the local extrema for .
is a local maxima
is a local minima
Step 15