What are the points of inflection, if any, of #f(x)= x^5 -2 x^3 + 3 x^2-2 #?

Answer 1

I am using the graph to approximate the location of the point of

inflexion for x little less than #-0.5#,

#f''(x)=20x^3-12x^2+6; f''(0)=-2<0 and f''(-1)=2 >0#.
So, f'' = 0 for x close to #-0.5#. The value correct to a specified

Using a numerical method, the number of significant digits can be obtained.

iterative method, with starer as #-0.5#. If required, I would give this

graph{x^5-2x^3+3x^2-2 [-10, 10, -5, 5]} in the upcoming edition.

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Answer 2
To find the points of inflection of \( f(x) = x^5 - 2x^3 + 3x^2 - 2 \), we first need to find the second derivative of the function, then solve for where the second derivative equals zero and where it changes sign. First, find the second derivative: \( f''(x) = 20x^3 - 6x \) Now, set \( f''(x) = 0 \) and solve for \( x \): \( 20x^3 - 6x = 0 \) \( 2x(10x^2 - 3) = 0 \) \( x = 0 \) or \( 10x^2 - 3 = 0 \) For \( 10x^2 - 3 = 0 \): \( 10x^2 = 3 \) \( x^2 = \frac{3}{10} \) \( x = \pm \sqrt{\frac{3}{10}} \) So, the potential points of inflection are \( x = 0 \), \( x = \sqrt{\frac{3}{10}} \), and \( x = -\sqrt{\frac{3}{10}} \). Now, we need to check the sign of the second derivative in intervals determined by these points. For \( x < -\sqrt{\frac{3}{10}} \), \( f''(x) > 0 \), indicating concavity is upward. For \( -\sqrt{\frac{3}{10}} < x < 0 \), \( f''(x) < 0 \), indicating concavity is downward. For \( 0 < x < \sqrt{\frac{3}{10}} \), \( f''(x) < 0 \), indicating concavity is downward. For \( x > \sqrt{\frac{3}{10}} \), \( f''(x) > 0 \), indicating concavity is upward. So, \( x = \sqrt{\frac{3}{10}} \) and \( x = -\sqrt{\frac{3}{10}} \) are the points of inflection.
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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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