How do you find the value of k for which #f(x)=x^35x^2+3x+k# has 11 as its relative minimum?
The equation's roots are the critical points:
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To find the value of ( k ) for which ( f(x) = x^3  5x^2 + 3x + k ) has ( 11 ) as its relative minimum, you need to determine the critical points of the function and then use the second derivative test.

Find the first derivative of ( f(x) ) and set it equal to zero to find critical points. [ f'(x) = 3x^2  10x + 3 ] Setting ( f'(x) = 0 ) gives: [ 3x^2  10x + 3 = 0 ] Solving this quadratic equation will give you the critical points.

Once you have the critical points, use the second derivative test to determine whether each critical point corresponds to a relative minimum, maximum, or neither. [ f''(x) = 6x  10 ] Evaluate ( f''(x) ) at each critical point.

Since you're looking for the value of ( k ) where the function has a relative minimum of ( 11 ), set ( f(x) = 11 ) and solve for ( x ). Then substitute the found ( x ) into ( f(x) ) to find the corresponding ( k ).
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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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