Finding the Constant for a Relative Minimum of 11 for a Cubic Function

[drandhawa]

Homework Statement

Let f be a function given by f(x)= x^3 - 5x^2 + 3x + K, where k is a constant. Find the value of k for which f has 11 as it's relative minimum.

Homework Equations

f(x)= x^3 - 5x^2 + 3x + K

The Attempt at a Solution

First, I found the derivative of f(x), found the critcal numbers, and then tested for relative minimum. There was a relative minimum at x=3. Now I'm not quite sure what to do so that 11 is a relative min.

 

[HallsofIvy]

Well, if the minimum is AT x= 3, what is the value of that minimum? That is, what is f(3)?

 

[drandhawa]

I plugged in 3 for f(x) and got -9 + k = f(x)

 

[drandhawa]

what di I do about the K?

 

[lax1113]

drand,
Perhaps the point of the k is that you plug in a (K)onstant that will make the function equal to what you have listed. So if x=3 is the x-coordinate of the minimum, and you are saying it needs to be 11, what value could you put in for k that will make the equation true for both criteria?

 

[drandhawa]

Oh, so because at the min pt is at (3,-9), I just have to shift the graph up so that the min pt would be (3,11). This means that the k value would have to be 20, right?

 

[HallsofIvy]

No, the minimum point is NOT at (3, -9). It is at (3, -9+ k) and you want -9+ k= 11. Just solve that equation. (You get the same answer of course.)