Calculus Examples

Find the second derivative.
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Find the first derivative.
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By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Evaluate .
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Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by to get .
Find the second derivative.
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By the Sum Rule, the derivative of with respect to is .
Evaluate .
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Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by to get .
Evaluate .
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Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by to get .
The derivative of with respect to is .
Set the second derivative equal to then solve the equation .
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Factor out of .
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Factor out of .
Factor out of .
Factor out of .
Divide each term by and simplify.
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Divide each term in by .
Reduce the expression by cancelling the common factors.
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Cancel the common factor.
Divide by to get .
Divide by to get .
Set equal to and solve for .
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Set the factor equal to .
Add to both sides of the equation.
The solution is the result of and .
Find the points where the second derivative is .
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Substitute in to find the value of .
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Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
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Remove parentheses around .
Raising to any positive power yields .
Remove parentheses around .
Raising to any positive power yields .
Multiply by to get .
Add and to get .
The final answer is .
The point found by substituting in is . This point can be an inflection point.
Substitute in to find the value of .
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Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
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Remove parentheses around .
Raise to the power of to get .
Remove parentheses around .
Raise to the power of to get .
Multiply by to get .
Subtract from to get .
The final answer is .
The point found by substituting in is . This point can be an inflection point.
Determine the points that could be inflection points.
Split into intervals around the points that could potentially be inflection points.
The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
Substitute any number from the interval into the second derivative and evaluate.
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Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
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Raise to the power of to get .
Multiply by to get .
Multiply by to get .
Add and to get .
The final answer is .
Find the decimal value of .
The graph is concave up on the interval because is positive.
Concave up on since
Substitute any number from the interval into the second derivative and evaluate.
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Substitute any number from the interval into the second derivative and evaluate.
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Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
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Raise to the power of to get .
Multiply by to get .
Multiply by to get .
Add and to get .
The final answer is .
Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
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Remove parentheses around .
One to any power is one.
Multiply by to get .
Multiply by to get .
Subtract from to get .
The final answer is .
The graph is concave down on the interval because is negative.
Concave down on since
Substitute any number from the interval into the second derivative and evaluate.
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Substitute any number from the interval into the second derivative and evaluate.
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Substitute any number from the interval into the second derivative and evaluate.
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Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
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Raise to the power of to get .
Multiply by to get .
Multiply by to get .
Add and to get .
The final answer is .
Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
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Remove parentheses around .
One to any power is one.
Multiply by to get .
Multiply by to get .
Subtract from to get .
The final answer is .
Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
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Remove parentheses around .
Raise to the power of to get .
Multiply by to get .
Multiply by to get .
Subtract from to get .
The final answer is .
Find the decimal value of .
The graph is concave up on the interval because is positive.
Concave up on since
The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.
Concave up on since
Concave down on since
Concave up on since
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