How do you find the critical numbers of # f(t)=t(4t)^(1/2)#?
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To find the critical numbers of the function ( f(t) = t(4  t)^{\frac{1}{2}} ), we need to find the values of ( t ) where the derivative of the function equals zero or is undefined.

Find the derivative of ( f(t) ) using the product rule. [ f'(t) = (4  t)^{\frac{1}{2}} + t \left( \frac{1}{2}(4  t)^{\frac{1}{2}} (1) \right) ] Simplify this expression.

Set the derivative equal to zero and solve for ( t ). [ (4  t)^{\frac{1}{2}} + t \left( \frac{1}{2}(4  t)^{\frac{1}{2}} (1) \right) = 0 ]

Solve for ( t ) in the above equation.

Check for any critical numbers by also considering where the derivative is undefined. In this case, the function is defined for all real numbers, so there are no points where the derivative is undefined.

The values of ( t ) obtained in step 3 are the critical numbers of the function ( f(t) ).
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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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