How do you find the max or minimum of #f(x)=3-x^2-6x#?

Answer 1

At #(-3, 12)#the function has a maximum.

Given -

#y=3-x^2-6x#

Determine the initial derivative.

#dy/dx=-2x-6#
Find for what value of #x; dy/dx# becomes zero
#dy/dx=0 => -2x-6=0#
#-2x-6=0# #-2x=6# #x=6/(-2)=-3#
Its interpretation is when #x# takes the value #0#; the slope of the curve is zero. The curve turns.
To find the point of turn, substitute #x=-3# in the given function
At #x=-3#
#y=3-(-3)^2-6(-3)# #y=3-9+18=12#
At #(-3, 12)#the curve's slope is zero. The curve truns.

It could be the maximum or minimum point.

Compute the second derivative to determine this.

#(d^2y)/(dx^2)=-2 <0#

There is a maximum to the curve because the second derivative is less than zero.

At #(-3, 12)#the function has a maximum.

graph{3-x^2-6x [-16.24, 16.24, -16.48, 32.48]}

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Answer 2

To find the maximum or minimum of ( f(x) = 3 - x^2 - 6x ), you need to differentiate the function with respect to ( x ) and set the derivative equal to zero. Then solve for ( x ) to find the critical points. Once you have the critical points, you can determine whether they correspond to a maximum or minimum by examining the sign of the second derivative.

First, differentiate ( f(x) ) with respect to ( x ) to find its derivative ( f'(x) ): [ f'(x) = -2x - 6 ]

Now, set ( f'(x) ) equal to zero and solve for ( x ) to find the critical points: [ -2x - 6 = 0 ] [ -2x = 6 ] [ x = -3 ]

Next, find the second derivative ( f''(x) ) to determine the concavity: [ f''(x) = -2 ]

Since ( f''(-3) = -2 ) is negative, the function is concave down at ( x = -3 ), indicating a maximum.

Therefore, the maximum of ( f(x) ) occurs at ( x = -3 ). To find the corresponding maximum value of ( f(x) ), plug ( x = -3 ) back into the original function: [ f(-3) = 3 - (-3)^2 - 6(-3) ] [ f(-3) = 3 - 9 + 18 ] [ f(-3) = 12 ]

So, the maximum value of ( f(x) ) is ( 12 ) at ( x = -3 ).

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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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