Final answer:
Statements a) g(-5) = -5 and b) f'(7) = 1/3 are true. Statement c) g(3) = -5 cannot be determined. Statement d) f''(-5) = 0 may or may not be false.
Step-by-step explanation:
To determine which of the statements must be false, we need to analyze the given information about f and g. We know that f(-5) = 7, which means that when we input -5 into function f, we get an output of 7. Similarly, we know that g'(7) = 3, which means the derivative of function g at 7 is 3. Now let's evaluate each statement:
a) g(-5) = -5: Since f(-5) = 7, and g is the inverse of f, this statement must be true. b) f'(7) = 1/3: This statement is true since the derivative of f is just the reciprocal of the derivative of g. Since g'(7) = 3, 1/3 is the correct value of f'(7). c) g(3) = -5: We do not have enough information to determine whether this statement is true or false. d) f''(-5) = 0: Since f is differentiable for all x, its second derivative exists. However, we cannot determine the value of f''(-5) based on the given information. Therefore, the statement d) f''(-5) = 0 may or may not be false.