Homework Soursecan someone help me with these true or false questions 1True
can someone help me with these true or false questions?
1.True / False. Every function f that is differentiable on the closed interval [a,b] is itself the derivative of some function g on the same closed interval [a,b]. Explain your reasoning if it is true or give a counterexample if it is false.
2.True/False. If f(x) is an even function and g(x) is even function, then (f+g)(x) is an even function. Explain your reasoning if it is true or give a counterexample if it is false.
3.True/False. If f is differentiable at c, then |f| is differentiable at c. Explain your reasoning if you answered true. Give a counterexample if you answered false.
1.True / False. Every function f that is differentiable on the closed interval [a,b] is itself the derivative of some function g on the same closed interval [a,b]. Explain your reasoning if it is true or give a counterexample if it is false.
2.True/False. If f(x) is an even function and g(x) is even function, then (f+g)(x) is an even function. Explain your reasoning if it is true or give a counterexample if it is false.
3.True/False. If f is differentiable at c, then |f| is differentiable at c. Explain your reasoning if you answered true. Give a counterexample if you answered false.
Solution
1. True. A differentiable function is continuous and therefore integrable.
2. True. If f(x) and g(x) are even, then f(-x) = f(x) and g(-x) = g(x). Then (f+g)(-x) = f(-x) + g(-x). Because f and g are even, this equals f(x) + g(x) = (f + g)(x) and we see f + g is even also.
3. False. y = x is differentiable at 0, but y = |x| is not.
![can someone help me with these true or false questions? 1.True / False. Every function f that is differentiable on the closed interval [a,b] is itself the deriv](/WebImages/47/can-someone-help-me-with-these-true-or-false-questions-1true-1147930-1761617452-0.webp "can someone help me with these true or false questions? 1.True / False. Every function f that is differentiable on the closed interval [a,b] is itself the deriv")
