Suppose that f'(x) > M for all x E [a, b]. Prove that f(b) > f(a) + M(b

Q: Suppose that g is a continuous function. Find the general solution of y" – y = g(t).

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Q: Let f be a function such that f' is continuous on [a, b] Show that a

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Q: Evaluate the function f(x) = e* + In (x + 7) at c, where –7 < c < 6.

A: Given information : f(x)=ex+ln(x+7) 'c' lies between -7 and 6

Q: 6. Let f be continuous on [a, b]. Show ffdF = f(x) [F(b) – F(a)] for some x € [a, b].

A: This is a consequence of the intermediate value theorem

Q: Let Ax) = 2x - 7x +4x. %3D a. On what interval(s) is f'(x) > 0? b. On what interval(s) is fx)…

A: Given: fx=2x3-7x2+4x   The Power law: ddxxn=nxn-1

Q: Suppose f and g are positive functions. Is the following true or false? Vf,g:N → R f(x) E O(g(x)) =…

A: Big-Θ notation: A function g(n) is in Θ(f(n)) if there exist constants c1, c2, and N such that…

Q: Let f(x) = ln x and g(x) Show that f(x) and g(x) are both increasing when x > 0. Vr.

A: Given, f(x) = lnx When f(x) is increasing then f'(x) &gt; 0  Differentiating f(x) with respect to x…

Q: the following statement. 'If yes, explain why. If no, provide a counterexample. For every real…

A: If the function is injective , then the above statement is true.

Q: Suppose f: X→Y, and A, B CY. Prove that (a) f-¹(AUB) = ƒ−¹(A) U ƒ−¹(B). (b) f-¹(ANB) = f¯¹(A) n…

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Q: Exercise 5: Compute / f(x)g" (x) – g(x)f"(x) dx.

A: Write the integration using two parts

Q: Suppose that the function f: [0, 1]  in R is continuous and thatf (x)&gt;= 2     if 0 &lt;= x &lt;…

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Q: Let f: Z → Z be defined by f(x) = x² + 1, and let C = {1,2,3}.

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Q: Suppose f: R → R is a function defined by f(x) = x³. (a) Find f([-1,1]) and f-¹([0, 8]). (b) Show…

A: We shall use the intermediate value theorem to answer this question

Q: Let f(x) = 4 – a?, a > 0 f- (x) =

A: Given:- f(x) =4-x2  To find:- f-1(x)

Q: Let f: X --> Y be a function, and let U be a subset of X.  (1) Prove or disprove that f maps X \ U…

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Q: Suppose that the function f: [0, 1] in R is continuous and thatf(x) &gt; 2      if 0&lt;=X &lt; 1.Is…

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Q: Suppose f(z) is analytic for |z| &lt;= 1, and |f(z)| &gt;=1. If f(0) =1, prove that f(z) is…

A: To prove that the given conditions on the analytic function force it to be the constant function…

Q: (2) Let f(x) = ax³ + bx? + cr + d, where a, b, c, d e R and a + 0. Prove that f(x) must have a root.…

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Q: verify 'If yes, explain why. If no, provide a counterexample. the following statement. For every…

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