What are the points of inflection, if any, of #f(x)=4x^3+15x^2-150x+4 #?

Answer 1

Inflection point is at #(-5/4, 1657/8) = (-1.25, 207.125)#

Given: #f(x) = 4x^3 + 15x^2 - 150x + 4#
Points of inflection are found when #f''(x) = 0#. First find the first derivative:
#f'(x) = 12x^2 + 30x - 150#
Find the second derivative: #f''(x) = 24x + 30#
Find inflections, #f''(x) = 0#:
#24x = -30#
#x = -30/24 = -5/4#
#f(-5/4) = 4(-5/4)^3 + 15(-5/4)^2 - 150(-5/4) + 4 #
#= 1657/8 = 207.125#
Inflection point is at #(-5/4, 1657/8) = (-1.25, 207.125)#
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Answer 2

To find the points of inflection, you need to first find the second derivative of the function, f(x). Then, solve for the values of x where the second derivative equals zero or is undefined. After obtaining these x-values, plug them into the original function to find the corresponding y-values. The points (x, y) obtained will be the points of inflection, if any.

  1. First derivative of f(x): f'(x) = 12x^2 + 30x - 150

  2. Second derivative of f(x): f''(x) = 24x + 30

  3. Set f''(x) = 0 and solve for x: 24x + 30 = 0 x = -30/24 = -5/4

  4. Plug x = -5/4 into the original function to find the corresponding y-value: f(-5/4) = 4(-5/4)^3 + 15(-5/4)^2 - 150(-5/4) + 4

  5. Calculate f(-5/4) to find the y-coordinate of the point of inflection, if any.

The point of inflection, if it exists, would be (-5/4, y).

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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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