Math  /  Calculus

Question349. 9709 m22_qp_12 Q: 11
It is given that a curve has equation y=k(3xk)1+3xy=k(3 x-k)^{-1}+3 x, where kk is a constant. (a) Find, in terms of kk, the values of xx at which there is a stationary point.

Studdy Solution

STEP 1

What is this asking? We need to find the x-values where the slope of the curve y=k(3xk)1+3xy = k(3x - k)^{-1} + 3x is **zero**, meaning the curve is momentarily flat.
These x-values represent the **stationary points**. Watch out! Don't forget to consider what happens if 3xk=03x - k = 0, as this could cause problems in our calculations!
Also, remember a stationary point can be a maximum, minimum, or a point of inflection.

STEP 2

1. Rewrite the equation
2. Find the derivative
3. Set the derivative to zero and solve for *x*

STEP 3

Let's **rewrite** the equation y=k(3xk)1+3xy = k(3x - k)^{-1} + 3x in a more convenient form for taking the derivative.
Remember, (3xk)1(3x - k)^{-1} is the same as 13xk\frac{1}{3x - k}.
So, we can rewrite the equation as: y=k3xk+3x y = \frac{k}{3x - k} + 3x This makes it easier to see how to apply the power rule for differentiation in the next step!

STEP 4

Now, let's **find** the derivative of yy with respect to xx, which we denote as dydx\frac{dy}{dx}.
We'll use the power rule, which states that the derivative of xnx^n is nxn1nx^{n-1}.

STEP 5

The derivative of k3xk\frac{k}{3x - k} with respect to xx can be found using the chain rule.
Think of it as k(3xk)1k(3x - k)^{-1}.
The derivative is k(3xk)23-k(3x - k)^{-2} \cdot 3, which simplifies to 3k(3xk)2\frac{-3k}{(3x - k)^2}.

STEP 6

The derivative of 3x3x with respect to xx is simply **3**.

STEP 7

Putting it all together, we get: dydx=3k(3xk)2+3 \frac{dy}{dx} = \frac{-3k}{(3x - k)^2} + 3 This tells us how the slope of the curve changes as xx changes!

STEP 8

To find the **stationary points**, we need to find the xx values where the derivative is **zero**.
So, we set dydx=0\frac{dy}{dx} = 0 and solve for xx: 3k(3xk)2+3=0 \frac{-3k}{(3x - k)^2} + 3 = 0

STEP 9

Subtract 3 from both sides: 3k(3xk)2=3 \frac{-3k}{(3x - k)^2} = -3

STEP 10

Divide both sides by -3: k(3xk)2=1 \frac{k}{(3x - k)^2} = 1

STEP 11

Multiply both sides by (3xk)2(3x - k)^2: k=(3xk)2 k = (3x - k)^2

STEP 12

Take the square root of both sides: ±k=3xk \pm\sqrt{k} = 3x - k

STEP 13

Add kk to both sides: k±k=3x k \pm\sqrt{k} = 3x

STEP 14

Finally, divide both sides by 3 to isolate xx: x=k±k3 x = \frac{k \pm \sqrt{k}}{3} These are the xx values at the stationary points!

STEP 15

The xx values at which there are stationary points are x=k+k3x = \frac{k + \sqrt{k}}{3} and x=kk3x = \frac{k - \sqrt{k}}{3}.

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