Answer:
As x increases, the rate of change of g exceeds the rate of change of f.
Explanation:
Given
![f(x) = 90x^2 + 180x + 92](https://img.qammunity.org/2022/formulas/mathematics/college/ae2liiwer85nxkgs7uuhk9fz1b9pldmd6h.png)
![\begin{array}{ccccccc}x & {0} & {1} & {2} & {3} & {4} & {5} & f(x) & {92} & {362} & {812} & {1442} & {2252} & {3242} \ \end{array}](https://img.qammunity.org/2022/formulas/mathematics/college/1my1f64vfme10rkqqx0l3etg9ooo16fe4b.png)
![g(x) = 6^x](https://img.qammunity.org/2022/formulas/mathematics/college/hwy5kp77god2yx8jhzz9edoztrmqtg3tdf.png)
![\begin{array}{ccccccc}x & {0} & {1} & {2} & {3} & {4} & {5} & g(x) & {1} & {6} & {36} & {216} & {1296} & {7776} \ \end{array}](https://img.qammunity.org/2022/formulas/mathematics/college/9xrl1005r9q7hh2h5dr509eqfrue4oj53b.png)
Required
Which of the options is true?
A. At
, f(x) has the same rate of change as g(x)
Rate of change is calculated as:
![m = (y_2 - y_1)/(x_2 - x_1)](https://img.qammunity.org/2022/formulas/mathematics/college/i1pa2mybgkt3dkd7j6tklhm5t9tvmo4g5v.png)
For f(x)
![f(x) = 90x^2 + 180x + 92](https://img.qammunity.org/2022/formulas/mathematics/college/ae2liiwer85nxkgs7uuhk9fz1b9pldmd6h.png)
![f(4.39) = 90*4.39^2 + 180*4.39 + 92 = 2616.689](https://img.qammunity.org/2022/formulas/mathematics/college/ygmc4ldg5ns711cjm8fuh9s2fsz795jpxh.png)
So, the rate of change is:
![m = (2616.689)/(4.39) = 596.06](https://img.qammunity.org/2022/formulas/mathematics/college/9ghxxma7gjf9xcp9mbxykrxmzi0sato7at.png)
For g(x)
![g(x) = 6^x](https://img.qammunity.org/2022/formulas/mathematics/college/hwy5kp77god2yx8jhzz9edoztrmqtg3tdf.png)
![g(4.39) = 6^(4.39) = 2606.66](https://img.qammunity.org/2022/formulas/mathematics/college/51peauj7v3m8bwz2by876phwp08qthie4i.png)
So, the rate of change is:
![m = (2606.66)/(4.39) = 593.77](https://img.qammunity.org/2022/formulas/mathematics/college/85e09j9r44bbudpfrj8fwa1lmm922gwft4.png)
The rate of change of both functions are not equal at x = 4.39. Hence, (a) is false.
B. Rate of change of g(x) is greater than f(x) with increment in x
Using the formula in (a), we have:
![\begin{array}{ccccccc}x & {0} & {1} & {2} & {3} & {4} & {5} & f(x) & {92} & {362} & {812} & {1442} & {2252} & {3242} & m &\infty & 362 & 406 & 480 & 563 &648.4\ \end{array}](https://img.qammunity.org/2022/formulas/mathematics/college/930of1k80qteihghe4o04i68tbgttxn2tu.png)
![\begin{array}{ccccccc}x & {0} & {1} & {2} & {3} & {4} & {5} & g(x) & {1} & {6} & {36} & {216} & {1296} & {7776} & m & \infty & 6 & 18 & 72 & 324 & 1555 \ \end{array}](https://img.qammunity.org/2022/formulas/mathematics/college/n190l44aoe16ltj6aq2pmbm0adwuar6qds.png)
From x = 1 to 4, the rate of change of f is greater than the rate of g.
However, from x = 5, the rate of change of g is greater than the rate of f.
This means that (b) is true.
The above table further shows that (c) and (d) are false.