Question

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 1

To find a possible value for g(6), we can use the Mean Value Theorem (MVT) given that g'(x) > 0 and g''(x) > 0 for all real numbers x. From the given information about g(4) and g(5), we can determine that g(x) is a strictly increasing and concave up function, and thus, the value of g(x) will continue to increase as x increases. Therefore, among the given options, the possible value for g(6) is 27.

Step-by-step explanation:

To find a possible value for g(6), we can use the Mean Value Theorem (MVT). Since g'(x) > 0 for all real numbers x and g''(x) > 0 for all real numbers x, this means that g(x) is a strictly increasing and concave up function.

Since g(4) = 12 and g(5) = 18, we know that the function is increasing and has a positive slope. Therefore, the value of g(x) will continue to increase as x increases.

Therefore, of the given options, the possible value for g(6) that is greater than g(5) is 27 (option e).

Answer 2

The function g is both increasing and concave up. Since g(5)=18 and g must increase at a rate that is at least as much as the previous interval (which was from 12 to 18, a change of 6), therefore g(6) must be at least 24. The only option greater than 24 is 27.

Step-by-step explanation:

We are given that g is a twice differentiable function with g'(x) > 0 and g''(x) > 0 for all real numbers x, which indicates that g is an increasing and concave-up function. Given g(4) = 12 and g(5) = 18, we want to find a possible value for g(6).

Since g'(x) > 0, g is increasing, meaning that g(6) must be greater than g(5) = 18. Furthermore, because g''(x) > 0, g is concave up, meaning that the rate of increase of g is also increasing. Thus, the increase from g(5) to g(6) should be at least as much as the increase from g(4) to g(5), which is 6 (from 12 to 18).

Therefore, g(6) must be greater than g(5) + 6 = 18 + 6 = 24. The only possible value provided in the options that is greater than 24 is 27. Hence, the answer is e. 27.