How do you find all critical point and determine the min, max and inflection given #f(x)=x^3+x^2-x#?
Critical Points are:
We have To identify the critical vales, we differentiate and find find values of Differentiating wrt At a critical point, Ton find the y-coordinate we substitute the required value into So the critical points are We identify the nature of these critical points by looking at the sign of second derivative, and Differentiating [1] wrt
Incidental, As this is a cubic with a positive coefficient of
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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^2-x [-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:
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Find the first derivative: [ f'(x) = \frac{{d}}{{dx}}(x^3 + x^2 - x) ]
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Set ( f'(x) ) equal to zero and solve for ( x ) to find critical points.
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After finding critical points, find the second derivative: [ f''(x) = \frac{{d^2}}{{dx^2}}(x^3 + x^2 - x) ]
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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 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.
- What are the points of inflection, if any, of #f(x)=2x^3+x^2+x+3 #?
- How do you find the inflection points of the graph of the function: #y=x^3-15x^2+33x+100#?
- Is #f(x)=sinx# concave or convex at #x=pi/5#?
- How do you use the first and second derivatives to sketch #y = x / (x^2 - 9)#?
- What is the second derivative of #f(x)= e^sqrt(3x-7)#?

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