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The above calculator is an online tool which shows output for the given input. This calculator, which makes calculations very simple and interesting. If an input is given then it can easily show the result for the given number. Calculus ExamplesFind the Local Maxima and Minima -(x+1)(x-1)^2 Step 2 Find the first derivative of the function. Expand using the FOIL Method. Apply the distributive property. Apply the distributive property. Apply the distributive property. Simplify and combine like terms. Since is constant with respect to , the derivative of with respect to is . Differentiate using the Product Rule which states that is where and . By the Sum Rule, the derivative of with respect to is . Differentiate using the Power Rule which states that is where . Since is constant with respect to , the derivative of with respect to is . Differentiate using the Power Rule which states that is where . Since is constant with respect to , the derivative of with respect to is . By the Sum Rule, the derivative of with respect to is . Differentiate using the Power Rule which states that is where . Since is constant with respect to , the derivative of with respect to is . Apply the distributive property. Apply the distributive property. Apply the distributive property. Apply the distributive property. Apply the distributive property. Use the power rule to combine exponents. Step 3 Find the second derivative of the function. By the Sum Rule, the derivative of with respect to is . Since is constant with respect to , the derivative of with respect to is . Differentiate using the Power Rule which states that is where . Since is constant with respect to , the derivative of with respect to is . Differentiate using the Power Rule which states that is where . Differentiate using the Constant Rule. Since is constant with respect to , the derivative of with respect to is . Step 4 To find the local maximum and minimum values of the function, set the derivative equal to and solve. Step 5 Find the first derivative. Find the first derivative. Expand using the FOIL Method. Apply the distributive property. Apply the distributive property. Apply the distributive property. Simplify and combine like terms. Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Product Rule which states that is where and . By the Sum Rule, the derivative of with respect to is . Differentiate using the Power Rule which states that is where . Since is constant with respect to , the derivative of with respect to is . Differentiate using the Power Rule which states that is where . Since is constant with respect to , the derivative of with respect to is . By the Sum Rule, the derivative of with respect to is . Differentiate using the Power Rule which states that is where . Since is constant with respect to , the derivative of with respect to is . Apply the distributive property. Apply the distributive property. Apply the distributive property. Apply the distributive property. Apply the distributive property. Use the power rule to combine exponents. The first derivative of with respect to is . Step 6 Set the first derivative equal to then solve the equation . Set the first derivative equal to . Factor the left side of the equation. For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is . Apply the distributive property. Factor out the greatest common factor from each group. Group the first two terms and the last two terms. Factor out the greatest common factor (GCF) from each group. Factor the polynomial by factoring out the greatest common factor, . Remove unnecessary parentheses. If any individual factor on the left side of the equation is equal to , the entire expression will be equal to . Set equal to and solve for . Subtract from both sides of the equation. Divide each term in by and simplify. Cancel the common factor of . Cancel the common factor. Move the negative in front of the fraction. Set equal to and solve for .
Add to both sides of the equation. The final solution is all the values that make true. Step 7 Find the values where the derivative is undefined. 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. Step 8 Critical points to evaluate. Step 9 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 10 Evaluate the second derivative. Cancel the common factor of . Move the leading negative in into the numerator. Cancel the common factor. Step 11 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 12 Replace the variable with in the expression. Write as a fraction with a common denominator. Combine the numerators over the common denominator. To write as a fraction with a common denominator, multiply by . Combine the numerators over the common denominator. Move the negative in front of the fraction. Use the power rule to distribute the exponent. Apply the product rule to . Apply the product rule to . Multiply by by adding the exponents. Use the power rule to combine exponents. Step 13 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 14 Evaluate the second derivative. Step 15 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 16 Replace the variable with in the expression. Raising to any positive power yields . Step 17 These are the local extrema for . is a local minima is a local maxima How do you find the local maximum and minimum?To find the local maxima and minima of a function f on an interval [a,b]: Solve f′(x)=0 to find critical points of f. Drop from the list any critical points that aren't in the interval [a,b].
How do you find the local maximum?To find the local maximum, we must find where the derivative of the function is equal to 0. Given that the derivative of the function yields using the power rule . We see the derivative is never zero. However, we are given a closed interval, and so we must proceed to check the endpoints.
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