Question

Multiple Choice If \(f\) is a continuous, decreasing function on \([0,10]\) with a critical point at \((4,2),\) which of the following statements must be false? (A) \(f(10)\) is an absolute minimum of \(f\) on \([0,10] .\) (B) \(f(4)\) is neither a relative maximum nor a relative minimum. (C) \(f^{\prime}(4)\) does not exist. (D) \(f^{\prime}(4)=0\) (E) \(f^{\prime}(4)<0\)

Short answer

The statement that must be false is (E) \(f^{\prime}(4)<0\).

Step-by-step solution

Evaluate statement (A)

The function is continuously decreasing on [0,10]. Thus, the function value at 10 will be the minimum among all points, i.e., f(10) is definitely an absolute minimum. So, statement (A) is true.

Evaluate statement (B)

Since the function is continuously decreasing on the interval, the value at any given point will always be higher than the value at any subsequent point. Hence, at the point (4,2), which is a critical point, f(4) cannot be either a relative maximum or a relative minimum, as the function doesn't switch its increase-decrease behaviour at this point. Statement (B) is also true.

Evaluate statement (C)

At critical points, the derivative does not exist or is equal to zero. However, we are given that f is a continuous function, therefore the derivative can exist at (4,2). Thus, it is possible that f'(4) exists, making statement (C) true.

Evaluate statement (D)

As already noted, at critical points, the derivative either does not exist or equals zero. Since f'(4) might exist (as mentioned in statement C), it could be that the derivative at the point equals zero, thus statement (D) is true.

Evaluate statement (E) and conclude

For a function to be decreasing, the derivative has to be less than zero. However, when the function reaches a critical point, the derivative can be zero or undefined. Thus, statement (E) which claims that f'(4)<0 must be false. The derivative at a critical point could be zero and truth of this statement contradicts with given condition that it is a critical point at (4,2).