For the function [math: f(x)] , if the point of inflection is located at [math: (a,b)] , find the value of [math: a] .[math: 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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The problem asks: For the function f(x) = x^3 − 5x^2 − 9x + 45, if the point of inflection is located at (a, b), find the value of a, correct to 2 decimal places.
First, recall that inflection points occur where the second derivative changes sign, i.e., where f''(x) = 0 (provided the concavity actually changes).
Next, compute derivatives:
- f'(x) = 3x^2 − 10x − 9
- f''(x) = 6x − 10
To locate potential inflection points, set f''(x) = 0:
6x − 10 = 0 ⇒ 6x = 10 ⇒ x = 10/6 = 5/3 ≈ 1.666666...
Thus the x-coordinate of the inflection point is a = 5/3.
Rounding to two decimal places gives a ≈ 1.67.
Notes on potential pitfalls: If you forget to compute the second derivative, you might try using the first derivative or looking for where f' has a local extremum, but those do not identify inflection points. Also, even if f''(x) = 0, you should verify that concavity changes around that x; for a cubic, this typically happens at the single root of f''(x).
Therefore, the value of a is 1.67 when rounded to two decimals.

UCIC 202503 EMTH118 Quiz 9

For the function [math: f(x)] , if the point of inflection is located at [math: (a,b)] , find the value of [math: a] .[math: 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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