We have to find out which of these statements are true. I narrowed it down to 2 answers.
a) if f"(0)=0, the the graph of f changes concavity at x=0
b) If the function f is continuous on the interval [a,b] and the integral from a to b of f(x)dx=0, then f must have at least on zero between a and b
a), yes true,
setting f''(x) = 0 gives you the point of inflection,
which is where the concavity changes
b) not true,
counterexample:
let f(x) = 2x^2 - 12x + 22
in the interval [1,8] the graph is entirely above the x-axis, and f'(x) = 4x - 12
f '(3) = 0
so I found a value of x for which the derivative is zero in the given interval but there is no root in the given interval
THANK YOU SO MUCH!
To determine which of these statements are true, let's analyze each statement separately.
Statement a) if f"(0) = 0, then the graph of f changes concavity at x = 0.
To understand this statement, we need to recall some concepts related to concavity and the second derivative of a function. The second derivative measures how the first derivative changes.
First, let's consider what it means for a function to change concavity. The concavity of a function determines whether the graph is cup-shaped (concave up) or bowl-shaped (concave down).
If the second derivative of a function changes sign at a point, it indicates a change in concavity. For example, if the second derivative is positive to the left of a point and negative to the right of that point, then the graph changes concavity at that point.
Therefore, for statement a) to be true, we need to determine if f"(0) = 0 implies a change in concavity at x = 0.
To confirm this, we will apply the Second Derivative Test. The Second Derivative Test states that:
- If f"(x) > 0 for x < c, and f"(x) < 0 for x > c, then f has a relative maximum at x = c.
- If f"(x) < 0 for x < c, and f"(x) > 0 for x > c, then f has a relative minimum at x = c.
- If f"(x) does not change sign at x = c, then there is no change in concavity.
If f"(0) = 0, it means that the second derivative is not positive nor negative at x = 0, implying that it does not change sign. Thus, statement a) is false. The graph of f does not change concavity at x = 0 if f"(0) = 0.
Now let's move on to statement b):
Statement b) If the function f is continuous on the interval [a, b] and the integral from a to b of f(x) dx = 0, then f must have at least one zero between a and b.
To analyze this statement, we will apply the Mean Value Theorem for Integrals. The Mean Value Theorem for Integrals states that if a function f is continuous on the interval [a, b], then there exists at least one number c in the interval (a, b) such that the integral from a to b of f(x) dx equals f(c) times the length of the interval (b - a).
Using this theorem, we can conclude that if the integral from a to b of f(x) dx = 0, it indicates that f(c) = 0 for at least one value of c in the interval (a, b). In other words, f has at least one zero between a and b.
Therefore, statement b) is true. If the function f is continuous on the interval [a, b] and the integral from a to b of f(x) dx = 0, then f must have at least one zero between a and b.
In summary:
a) Statement a) is false. The graph of f does not change concavity at x = 0 if f"(0) = 0.
b) Statement b) is true. If the function f is continuous on the interval [a, b] and the integral from a to b of f(x) dx = 0, then f must have at least one zero between a and b.