For the function f(x) f(x) , if the point of inflection is located at (a,b) (a,b) , find the value of a a .f(x)=x3+−5x2+−9x+45 f(x)=x^3+-5x^2+-9x+45 Calculate your answer correct to 2 decimal places.

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BlackTom AI · Accepted Answer

To locate the point of inflection, we need to examine the second derivative of f.
1) Start with the given function: f(x) = x^3 - 5x^2 - 9x + 45.
2) Differentiate to obtain f'(x): f'(x) = 3x^2 - 10x - 9.
3) Differentiate again to get the second derivative: f''(x) = 6x - 10.
4) Inflection points occur where f''(x) = 0 (provided the concavity changes). So solve 6x - 10 = 0 for x.
5) Solve: 6x = 10 ⇒ x = 10/6 = 5/3 ≈ 1.666... .
6) Since the question asks for the value of a (the x-coordinate of the inflection point) to two decimals, we round 1.666... to 1.67.
7) Therefore, the x-coordinate of the inflection point is a ≈ 1.67.

Note on reasoning: The inflection point is found where the concavity changes sign, which corresponds to f''(x) crossing zero. For a cubic with a nonzero leading coefficient, f'' is linear, so it has exactly one root, giving the unique x-coordinate of the inflection point as 5/3. The numerical value is then rounded to two decimals as required.

UCIC 202503 EMTH118 Quiz 9

For the function f(x) f(x) , if the point of inflection is located at (a,b) (a,b) , find the value of a a .f(x)=x3+−5x2+−9x+45 f(x)=x^3+-5x^2+-9x+45 Calculate your answer correct to 2 decimal places.

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