How do you find the critical points for #y=x+2x^1#?
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To find the critical points of ( y = x + 2x^{1} ), follow these steps:
 Find the first derivative of the function: ( y' = 1  2x^{2} ).
 Set the first derivative equal to zero and solve for ( x ): ( 1  2x^{2} = 0 ).
 Solve for ( x ) in the equation: ( 2x^{2} = 1 ).
 Take the reciprocal of both sides: ( x^2 = 2 ).
 Solve for ( x ): ( x = \pm \sqrt{2} ).
Therefore, the critical points of the function are ( x = \sqrt{2} ) and ( x = \sqrt{2} ).
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To find the critical points for the function ( y = x + 2x^{1} ), first find its derivative, set it equal to zero, and solve for ( x ). Then, check these values to ensure they are valid critical points.

Find the derivative of the function: [ y' = 1  2x^{2} ]

Set the derivative equal to zero: [ 1  2x^{2} = 0 ]

Solve for ( x ): [ 1 = 2x^{2} ] [ x^{2} = \frac{1}{2} ] [ x^2 = 2 ] [ x = \pm \sqrt{2} ]

Check these values to ensure they are valid critical points by plugging them into the original function or its derivative.
Therefore, the critical points are ( x = \sqrt{2} ) and ( x = \sqrt{2} ).
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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