Question

The function meets the following requirements: i)At x=0, the value of the function is 0, and the slope of the graph of the function is 0. ii)At x=4, the value of the function is 1, and the slope of the graph of the function is 1. iii) Between x=0 and x=4, the function is increasing.
a. Let f(x)=ax^2, where (a) is a nonzero constant. Show that it is not possible to find a value for a so that f meets requirement (ii) above.
b. Let g(x)=cx^3-x^2/16, where c is a nonzero constant. Find the value of c so that g meets requirement (ii) above. Show the work that leads to your answer.
c. Using the function g and your value of c from part (b), show that g does not meet requirement (iii) above.
d. Let h(x)=x^n/k, where k is nonzero constant and n is a positive integer. find the values of k and n so that h meets requirement (ii) above. Show that h also meets requirements (i) and (iii) above.

Answer 1

A. f(x)=ax^2
f(4)=16a=1
Meanwhile, f'(x)=2ax
f'(4)=8a=1
No single non-zero a satisfies (ii). So no such function exists.

Answer 2

thanks you so much! can you help me on b and d?

Answer 3

Before I help you, I want to make sure I'm working with the right functions:
B. \(\large g(x)=cx^3-\frac{1}{16}x^2 \text{ or } \frac{1}{16}(cx^3-x^2)\)
D. \(\large h(x)=x^{n/k} \text{ or } \frac{x^n}{k}\)
Tell me whether it's the 1st or the 2nd.