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.
x –8 –3 0 3 8
f ′(x) 5 4 0 –2 –4
If f ′(x) is always decreasing, which statement about f(x) must be true?
f(x) has a relative maximum at x = 0.
f(x) is concave upwards for all x.
f(x) has a point of inflection at x = 0.
f(x) passes through the origin.
To determine which statement about f(x) must be true if f ′(x) is always decreasing, we can analyze the given table of values for f ′(x).
From the table, we observe that as x increases, the corresponding values of f ′(x) decrease. This indicates that the derivative of the function f(x) is negative or decreasing.
Since the derivative of a function describes the rate at which the function is changing, a decreasing derivative implies that the function is decreasing at a decreasing rate. In other words, f(x) is a concave-down function.
Therefore, the statement that must be true about f(x) is that it is a concave-down function.
To determine which statement about f(x) must be true if f'(x) is always decreasing, we need to analyze the relationship between the behavior of f'(x) and f(x).
When f'(x) is decreasing, it means that the rate of change of f(x) with respect to x is getting smaller as x increases. This implies that f(x) is concave down.
Let's check the concavity of f(x) based on the given table:
- For x = -8, f'(x) = 5.
- For x = -3, f'(x) = 4.
- For x = 0, f'(x) = 0.
- For x = 3, f'(x) = -2.
- For x = 8, f'(x) = -4.
We can observe that the values of f'(x) are always decreasing. Now, if f'(x) is decreasing, it means the slope of f(x) is becoming less steep as x increases. Hence, f(x) is a concave-down function.
Therefore, the statement that must be true about f(x) is that it is a concave-down function.
d^2y/dx^2 is -
now let me see if I throw a ball up to the right (x +) at 45 degrees
the slope is + for a while
but it keeps decreasing :)
It is 0 at the top then gets negative