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

You first take the derivative

By signing up, you agree to our Terms of Service and Privacy Policy

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

- Find the critical points by setting the derivative of ( f(x) ) equal to zero and solve for ( x ).
- Determine if the critical points lie within the given interval ( x \geq 0 ).
- Evaluate ( f(x) ) at the critical points and endpoints of the interval.
- 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.

By signing up, you agree to our Terms of Service and Privacy Policy

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.

- Is #f(x)=(x+4)^2+x^2-3x # increasing or decreasing at #x=-2 #?
- Using the principle of the mean-value theorem on the indicated interval, how do you find all numbers c that satisfy the conclusion of the theorem #f(x) = x³ + 5x² - 2x - 5#; [-1, 2]?
- What are the values and types of the critical points, if any, of #f(x)=x^2-5x+4#?
- How do you verify that the function #f(x)= sin(22pix)# satisfies the three hypotheses of Rolle's Theorem on the given interval [-1/11, 1/11] and then find all numbers c that satisfy the conclusion of Rolle's Theorem?
- What are the extrema and saddle points of #f(x,y) = xy + 1/x^3 + 1/y^2#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7