Question
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.
(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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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: