How do you find all critical point and determine the min, max and inflection given #f(x)=x^3+x^2x#?
Critical Points are:
We have
To identify the critical vales, we differentiate and find find values of
Differentiating wrt
# f'(x) = 3x^2 + 2x  1 # .... [1]At a critical point,
# f'(x)=0 #
# f'(x)=0 => 3x^2 + 2x  1 = 0 #
# :. (3x1)(x+1) = 0 #
# x=1,1/3 # Ton find the ycoordinate we substitute the required value into
#f(x)#
# x=1 => f(1)=1+1(1)=1 #
# x=1/3 => f(2/3)=1/27+1/9+1/3=5/27 # So the critical points are
#(1,1)# and#(1/3,5/27)# We identify the nature of these critical points by looking at the sign of second derivative, and
# { ( f''(x) < 0, => f'(x) " is decreasing" => "maximum" ), ( f''(x) = 0, => f'(x) " is stationary" => "inflection" ), ( f''(x) > 0, => f'(x) " is increasing" => "minimum" ) :} # Differentiating [1] wrt
#x# gives s;
# f''(x) = 6x + 2 #
# x=1 => f''(2)=6+2 < 0 # , ie a maximum
# x=1/3 => f''(1)= 2+2>0# , ie a minimum
Incidental, As this is a cubic with a positive coefficient of
#x^3# , we can deduce that the critical point corresponding to the smallest value of#x# must be the maximum and that corresponding to the larger value must be the maximum. It is not possible to have any other possibility!
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The max is at
The min is at
The inflexion point is at
We have to calculate the first and second derivative.
So, we do a sign chart
graph{x^3+x^2x [2.43, 2.436, 1.217, 1.215]}
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To find all critical points of ( f(x) = x^3 + x^2  x ) and determine whether they correspond to a minimum, maximum, or point of inflection, follow these steps:
 Find the first derivative of ( f(x) ) and set it equal to zero to find critical points.
 Use the second derivative test to determine whether each critical point corresponds to a minimum, maximum, or point of inflection.
Let's proceed with the calculations:

Find the first derivative: [ f'(x) = \frac{{d}}{{dx}}(x^3 + x^2  x) ]

Set ( f'(x) ) equal to zero and solve for ( x ) to find critical points.

After finding critical points, find the second derivative: [ f''(x) = \frac{{d^2}}{{dx^2}}(x^3 + x^2  x) ]

Use the second derivative test:
 If ( f''(x) > 0 ), the critical point corresponds to a local minimum.
 If ( f''(x) < 0 ), the critical point corresponds to a local maximum.
 If ( f''(x) = 0 ), the test is inconclusive, and further analysis is needed.
Let's solve for critical points and apply the second derivative test to determine the nature of each critical point.
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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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