Answer:
The graph of f is always concave down ⇒ the first answer
Step-by-step explanation:
* Lets explain how to solve the problem
- Remember that : If f(x) is a function then the solutions to the
equation f′(x) = 0 gives the maximum and minimum values to f(x)
- The value of x gives maximum if f′′(x) is negative and minimum if
f′′(x) is positive.
- Inflection points of the function f(x) are found the solutions of the
equation f′′(x) = 0
* Lets solve the problem
- The graph of f'(x) is continuous means that the graph is unbroken line
- The graph of f'(x) decreasing with an x-intercept at x = 2 means
f'(2) = 0
- The differentiation of a function equal to zero at the critical point
(minimum or maximum) of the function
∵ f'(x) = 0 at x = 2
∴ The x-coordinate of the critical point of f(x) is 2
- If the differentiation of the function is decreasing, then the critical
point of the function is maximum point
∵ f'(x) is decreasing
∴ The critical point of the f(x) is maximum point
- That means the slope of curve is negative
∴ The graph of f is concave down at x = 2
* The right answer is the graph of f is always concave down
