Question
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The point is a stationary point on the graph of , where is a constant.
(i) Show that .
(ii) Determine the value of .
(iii) Find the coordinates of the other stationary point.

Ask by Sherman Olson. in Singapore
Jan 26,2025

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Tutor-Verified Answer

Answer

, , and the other stationary point is at .

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The Deep Dive

To solve for the constants and , and find the coordinates of the other stationary point, we follow these steps:

(i) Show that .

First, to find stationary points, we need to differentiate the function .
The derivative is:
At the stationary point , the derivative must equal zero:

(ii) Determine the value of .

Now we substitute and into the original function to find :

(iii) Find the coordinates of the other stationary point.

Substituting and back into the original function:
Setting the derivative equal to zero to find stationary points:
We can solve this quadratic equation using the quadratic formula where , , and :
This gives us:
We already have the stationary point . Now for the other stationary point:
Substituting back into the equation to find :
Converting everything to a common denominator of 27:
Thus, the coordinates of the other stationary point are:

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