Let g be a twice-differentiable function with g'(x) > 0 and g”(x) > 0 for all real numbers x, such that g(3) 12 and g(5) = 18. Which of 20, 21, and 22 are possible values for g(6)?
(A) 21 only
(B) 22 only
(C) 20 and 21 only
(D) 21 and 22 only
Answer:
g(6) must be greater than 21 which is 22 only.
Explanation:
What is function?
In mathematics, a function is an expression, rule, or law that establishes the relationship between an independent variable and a dependent variable.
Since g'(x) > 0 for all x and we know that g(x) is an increasing function.
Additionally, since g”(x) > 0 for all x, we know that g(x) is a concave-up function.
Since g(x) is increasing, we know that g(6) > g(5) = 18.
Now, we can use the fact that g(x) is concave-up to find a lower bound on g(6).
By the definition of concave-up, we know that the slope of the tangent line to g(x) is increasing.
This means that the slope of the line connecting (3,g(3)) to (5,g(5)) is less than the slope of the line connecting (5,g(5)) to (6,g(6)).
Using this information, we can find a lower bound on g(6):
g(5)- g(3) / (5-3) < g(6)- g(5)/ (6-5)
18- 12/2 < g(6)- 18
3 < g(6) – 18
g(6) > 21
Therefore, g(6) must be greater than 21.
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