What are the points of inflection of #f(x)= x^3 - 12x^2 + 2x + 15x #?

Answer 1

# "(4 , -30)" # is the inflection point.

Determining the points of inflection is by finding the second #" "# derivative then solve the equation for #f''(x)=0#. #" "# #" "# #f'(x) = 3x^2 - 24x + 2 + 15# #" "# #f''(x) = 6x -24# #" "# #" "# #f''(x) = 0# #" "# #6x -24 = 0# #" "# #rArr 6x = 24# #" "# #rArr x = 24/6 = 4# #" "# #" "# Finding its ordinate #f(4)#. #" "# #f(4) = 4^3 - 12(4)^2 + 2(4) + 15(4)# #" "# #f(4) = 64 - 192 + 8 + 90# #" "# #f(4) = -30# #" "# #" "#
Hence, # "(4 , -30)" # is the inflection point.
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Answer 2
To find the points of inflection of \( f(x) = x^3 - 12x^2 + 2x + 15x \), we first need to find its second derivative, then set it equal to zero and solve for \( x \). Finally, we'll use the second derivative test to determine whether these points are points of inflection. First derivative of \( f(x) \): \[ f'(x) = 3x^2 - 24x + 17 \] Second derivative of \( f(x) \): \[ f''(x) = 6x - 24 \] Setting \( f''(x) \) equal to zero: \[ 6x - 24 = 0 \] \[ x = 4 \] Second derivative test: For \( x < 4 \), \( f''(x) > 0 \), so \( f(x) \) is concave up. For \( x > 4 \), \( f''(x) > 0 \), so \( f(x) \) is concave up. Therefore, the only point of inflection is at \( x = 4 \).
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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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