Problem 14 Suppose that the function f: R rightarrow R has the property that Pro
ID: 2983848 • Letter: P
Question
Problem 14
Suppose that the function f: R rightarrow R has the property that Prove that f is differentiable at x=0 and that f'(0)=0. For real number a and b, define For what value of a and b is the function g: R rightarrow R differentiable at x=1? Suppose that the function g:R rightarrow R is differentiable at x=0, Also, suppose that for each natural number n, g(1/n)=0. Prove that g(0)=0 and g'(0)=0. Suppose that the function f: R rightarrow R is differentiable and monotonically increasing. Show that f'(x) 0 for all x. Suppose that the function f: R rightarrow R is differentiable and that there is a bounded sequence {xn} with Xn Xm, if n m, such that f(xn)=0 for every index n. Show that there is a point X0 at which f(X0)=0 and f'(X0)=0 (Hint: Use then Sequential Compactness Theorem.) Suppose that the function f: R rightarrow R is differentiable at X0. Analyze the limit [Hint: subtract and add f(X0) to the numerator.] suppose that the function f: R rightarrow R is differentiable at X0. Prove that Let the function f:R rightarrow R be differentiable at x = 0. Prove that Suppose that the function f: R rightarrow r is differentiable at 0. For real number a, b, and c, with c 0, prove thatExplanation / Answer
add and substract f(x0)
we have limit h->0 (f(x0+h)-f(x0))/h=f'(x0) and
limit -h->0 -f(x0-h)+f(x0)/h(note h->0 =>-h->0 also) i.e limit -h->0 f(x0-h)-f(x0)/-h which is inturn equal to f'(x0)
so answer=2f'(x0)