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Find the coordinates of the turning point on the curve with equation $y = 9 + 18x - 3x^2$ - Edexcel - GCSE Maths - Question 1 - 2021 - Paper 2

Question 1

Find the coordinates of the turning point on the curve with equation $y = 9 + 18x - 3x^2$. You must show all your working.

Worked Solution & Example Answer:Find the coordinates of the turning point on the curve with equation $y = 9 + 18x - 3x^2$ - Edexcel - GCSE Maths - Question 1 - 2021 - Paper 2

Step 1

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Answer

To find the turning points, we first need to differentiate the given equation with respect to $x$ . The derivative is found as follows:

$rac{dy}{dx} = rac{d}{dx}(9 + 18x - 3x^2) = 18 - 6x$

Step 2

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Answer

Next, we set the derivative equal to zero to find the critical points:

$18 - 6x = 0$

Solving for $x$ gives:

x = 3$$

Step 3

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Answer

To confirm that this point is indeed a turning point, we compute the second derivative:

$\frac{d^2y}{dx^2} = \frac{d}{dx}(18 - 6x) = -6$

Since the second derivative is negative, the point at $x = 3$ is a maximum, confirming a turning point.

Step 4

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Answer

Now we substitute $x = 3$ back into the original equation to find the corresponding $y$ value:

$y = 9 + 18(3) - 3(3^2) = 9 + 54 - 27 = 36$

Thus, the coordinates of the turning point are $(3, 36)$ .

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