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Question 8
The curve C has equation y = 2x^3 - 5x^2 - 4x + 2. (a) Find \( \frac{dy}{dx} \). (b) Using the result from part (a), find the coordinates of the turning points of... show full transcript
Step 1
Step 2
Answer
To find the turning points, we set ( \frac{dy}{dx} = 0 ):
Factoring gives:
which results in ( x = 2 ) and ( x = -\frac{1}{3} ).
Now, we substitute these x-values back into the original equation to find the corresponding y-coordinates:
For ( x = 2 ):
Thus, one turning point is ( (2, -10) ).
For ( x = -\frac{1}{3} ):
Calculate this value to find the second turning point.
Step 3
Step 4
Answer
To determine the nature of the turning points, we evaluate ( \frac{d^2y}{dx^2} ) at the turning points found earlier:
For ( x = 2 ):
( \frac{d^2y}{dx^2} = 12(2) - 10 = 24 - 10 = 14 ) (positive, indicating a local minimum).
For ( x = -\frac{1}{3} ):
( \frac{d^2y}{dx^2} = 12(-\frac{1}{3}) - 10 = -4 - 10 = -14 ) (negative, indicating a local maximum).
In summary, the turning point at ( (2, -10) ) is a minimum and the turning point at ( (-\frac{1}{3}, y) ) is a maximum.
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