How do you find the minimum values for #f(x)=2x^3-9x+5# for #x>=0#?

Answer 1

You first take the derivative

#f(x)=2x^3-9x+5->f'(x)=6x^2-9#
You have an extreme if #f'(x)=0#
#6x^2-9=0->6x^2=9->x^2=9/6=1.5#
#x=+-root 2 1.5=+-1.225...#
And we keep only the positive value. We put that in the formula to get #y#
Answer : #x~~1.225# and #y~~-2.358# graph{2x^3-9x+5 [-25.66, 25.68, -12.83, 12.81]}
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Answer 2

To find the minimum values of ( f(x) = 2x^3 - 9x + 5 ) for ( x \geq 0 ), follow these steps:

  1. Find the critical points by setting the derivative of ( f(x) ) equal to zero and solve for ( x ).
  2. Determine if the critical points lie within the given interval ( x \geq 0 ).
  3. Evaluate ( f(x) ) at the critical points and endpoints of the interval.
  4. The smallest value among these will be the minimum value of ( f(x) ) within the specified interval.

First, find the derivative of ( f(x) ) and set it equal to zero to find critical points:

( f'(x) = 6x^2 - 9 )

Set ( f'(x) = 0 ) and solve for ( x ):

( 6x^2 - 9 = 0 )

( x^2 = \frac{9}{6} = \frac{3}{2} )

( x = \pm \sqrt{\frac{3}{2}} )

However, since we're considering ( x \geq 0 ), we only need to consider ( x = \sqrt{\frac{3}{2}} ).

Next, evaluate ( f(x) ) at the critical point and endpoints of the interval:

( f(0) = 5 )

( f\left(\sqrt{\frac{3}{2}}\right) = 2\left(\sqrt{\frac{3}{2}}\right)^3 - 9\sqrt{\frac{3}{2}} + 5 )

Now, calculate ( f\left(\sqrt{\frac{3}{2}}\right) ) to find the minimum value.

After evaluating, compare the values of ( f(0) ) and ( f\left(\sqrt{\frac{3}{2}}\right) ) to find the minimum value.

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Answer from HIX Tutor

When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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