If f is decreasing and f(20)=10, which of the following must be incorrect? Explain.
a. (f^1)'(20)=-3
b. (f^1)'(10)=12
how would i go about figuring this out??? i'm not sure exactly what the concept in question is here.
Well if it is decreasing it does not have a positive slope.
To determine which statement must be incorrect, we need to understand the relationship between the given information and the options provided.
The fact that "f is decreasing" means that as the input (x) increases, the output (f(x)) decreases.
Now let's analyze the given information:
- f(20) = 10 means that the output of the function at x = 20 is 10.
Let's consider each option:
a. (f^1)'(20) = -3
This option represents the derivative of the function f at x = 20. If the derivative is negative (-3), it means that the function is decreasing at that point. Since we are given that f is decreasing at x = 20, this option could possibly be correct.
b. (f^1)'(10) = 12
This option represents the derivative of the function f at x = 10. If the derivative is positive (12), it means that the function is increasing at that point. However, the given information states that f is decreasing overall, so this option must be incorrect.
Therefore, the correct answer is b. (f^1)'(10) = 12 since it contradicts the given fact that f is decreasing.
To determine which statement is incorrect, you need to understand the concept of the derivative and its relationship to the behavior of a function.
The derivative, denoted as f'(x) or (df/dx), measures the rate at which a function changes at a specific point. In this case, f'(x) represents the rate of change of the function f(x).
Given that function f is decreasing, it means that as x increases, the corresponding y-values (or f(x) values) are decreasing. So, if f(20)=10, this implies that when x is 20, the function value is 10.
Now, let's analyze the given statements:
a. (f^1)'(20) = -3
This statement represents the derivative of f at x=20. Since f is decreasing, the derivative should be negative. So, it is possible that (f^1)'(20) equals -3. This statement might be correct.
b. (f^1)'(10) = 12
This statement represents the derivative of f at x=10. However, since f is decreasing, the derivative should be negative. If f(20)=10 and the function is decreasing, it suggests that when x decreases from 20 to 10, the function values should also decrease. Thus, a positive derivative value contradicts the fact that f is decreasing. Therefore, this statement is most likely incorrect.
To recap, the incorrect statement is b. (f^1)'(10) = 12, as the function f is decreasing and should have a negative derivative.