f is a function that is differentiable for all reals. The value of f ′(x) is given for several values of x in the table below.
The table:
x -8,-3,0,3,8
f'(x)-4,-2,0,4,5
If f ′(x) is always increasing, which statement about f(x) must be true?
f(x) passes through the origin
f(x) is concave down for all x
f(x) has a relative minimum at x=0
f(x) has an inflection point at x=0
My answer is the last one,d
f is a differentiable function on the interval [0,1], and g(x)=f(2x) the table below gives the values of f'(x). What is the value of g'(0.2)?
x .1,.2,.3,.4,.5
f'(x) 1,2,3,-4,5
Options:
2
-8
4 MY ANSWER
Cannot be determined
Come to think of it, I changed my mind.
g'(0.2) = f'(0.4)*2 = -8
Thank you!
To determine which statement about f(x) must be true if f'(x) is always increasing, we can analyze the given table. Since f'(x) is always increasing, the values are getting larger as x increases. From the table, we can see that f'(x) goes from -4 to -2 to 0 to 4 to 5. This means that the slope of the function is increasing, indicating that f(x) is becoming steeper as x increases.
Based on this information, we can conclude that f(x) is concave down for all x. So, the correct statement is: f(x) is concave down for all x.
For the second question, we are given a function f(x) and asked to find the value of g'(0.2) where g(x) = f(2x). To find g'(0.2), we need to find f'(x) at x = 0.2/2 = 0.1.
From the table, we see that f'(0.1) = 1. Since g(x) = f(2x) and we want g'(0.2), we need to evaluate f'(0.1) since 2x = 0.2 when x = 0.1.
Therefore, g'(0.2) is equal to f'(0.1), which is 1. So, the correct answer is 1.
#1 not D, since f' is always increasing, f" cannot be zero anywhere. So, no inflection points. I like C, since f'(0) = 0. f has a minimum, since f" > 0 and thus f is concave up.
g(x) = f(2x), so
g'(x) = f'(2x)*2
g'(0.2) = f'(0.2)*2 = 2*2 = 4
You are correct.