G(x) is twice differentiable where g(4)=12 and g(5)=18. g(6)=?

[lude1]

Homework Statement

Let g be a twice differentiable function with g'(x)>0 and g''(x)>0 for all real numbers x, such that g(4)=12 and g(5)=18. Of the following, which is a possible value for g(6)?

a. 15
b. 18
c. 21
d. 24
e. 27

Answer: e. 27

Homework Equations

The Attempt at a Solution

I guess the first question is the first line of the question. "Let g be a twice-differentiable function with g'(x)>0 and g''(x)>0" means the first and second derivative of g(x) is positive, right?

They gave me g(4)=12 and g(5)=18. Therefore, I can find the function by plugging it into y-y₁ = m(x-x₁, find m, and then find b.

12-18 = m(4-5)
-6 = m(-1)
m=6

12 = 6(4) + b
b = -12

y= 6x-12​

Since they want g(6), I plugged in 6 for x.

y= 6(6)-12
y= 24​

Though that is answer d, it is incorrect.

I guess my problem might come from the first sentence. They told me g'(x) and g''(x) is positive, but I don't know how that helps me.

 

[D H]

Almost there! Your solution y=24 is the straight line solution with slope=6. In other words, g'(x)=6. Since this is a straight line, the second derivative is zero. However, the problem tells you that g'(x) and g''(x) are positive. This means g(6) must be greater than 24. The only possible choice is (e) 27.

 

[Mark44]

g' and g'' give you some idea about the shape of the graph. If g'(x) > 0, the graph is increasing. If g''(x) > 0, that gives you information about the concavity of the graph, whether the graph is concave up or concave down.

 

[lude1]

g''(x)>0 means there must be a place where it is concave up, right?

y= 6x-12 is a straight line.
y'=6 is also a straight line
y'' does not exist? which means there is no concavity which means g''(x) is not greater than 0.

Therefore when I plug in 6, I get 24. But because of the conditions above, 24 CANNOT be the answer. And since y'' must be positive, the answers cannot be anything below 24, leaving me with the only answer choice left, (e) 27?

 

[Mark44]

If y = 6x - 12, then y' = 6, and y'' = 0, so yes, it does exist, and there is no concavity.

 

[esc1729]

Your function must be monotone increasing and concave for all x. So an exponential function comes into mind. As you have two constraints, try g(x) = a exp( bx). Solving this for your two x values will lead to a very simple result with indeed g(6) = 27.

Erich

 

[Mark44]

Your function must be monotone increasing and concave for all x. So an exponential function comes into mind.

Or not. This problem can be solved without having to solve for a specific function.

As you have two constraints, try g(x) = a exp( bx). Solving this for your two x values will lead to a very simple result with indeed g(6) = 27.

Erich

 

[esc1729]

Yeah, I've just realized. It's quite some time since I'm out of school and we had no multiple choice tests then ...

 

[Mark44]

No, I didn't either, but a problem like this is good for emphasizing concepts at a high level without getting bogged down in computations.