Prove that if (c, f(c)) is a point of inflection of the graph of f and f" exists in an open interval that contains c, then f"(c) = 0. [Hint: Apply the First Derivative Test and Fermat's Theorem to the function g =f'.]

Q: 1. Consider the function f(x) = x6 - x³. a. Determine the absolute extremum using the First…

A: The given function is fx=x6-x3. a. Find the absolute extremum using the first derivative test. b.…

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A: Given: f(x) = ln (9 - x 2) To determine: The domain of the function and prove that it is continuous…

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A: Explained below

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Q: 1. Given the graph of f', the derivative of f, locate all extrema of f. JUSTIFY. Minimum at x =…

A: To find the minima and maxima from the graph of f'(x)

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Q: Let f (x) = x² – 4. a. Find f'(x) using the formal limit definition of derivative. -

A: The formula of formal limit definition of derivative is  : lim h →0  [{f(x+h) - f(x)} / h ]

Q: Suppose that a function ƒ satisfies the following conditions for all real values of x and y: i) ƒ(x…

A: The given conditions are,

Q: The function f(x) is everywhere continuous and differentiable. Prove that if the function f(x) has…

A: Given that f(x) is continuous and differentiable everywhere and f(x) has two real roots.

Q: Prove that for all a, b = R, the function f(x) = 3x² - 4x³ + 6x² + ax + b has exactly one point…

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Q: Q1: If the function f(x) define as f: R→ R such that f(x) = (x+1, x>2 x²-1, x≤2 Show that the…

A: This is the question to check the differentiability of the function.

Q: If f(x) is a continuous function on [3,7] and F(x) is an anti-derivative of f(x) such that F(3)=7…

A: Given :f(x) is continuous function 3,7Fx is anti-derivative of fxF3=7 and F7=3To determine…

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Q: Consider the function f(x)=3x2−4x+10,  0≤x≤8f(x)=3x2-4x+10,  0≤x≤8. Use the derivatives to…

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Q: Let g (x) = f' (x) be a continuous function. If the derivative of the continuous function g (x)…

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Q: Let f be a differentiable function such that f(4) = 3 and f'(x) = -3x + 3. What is the approximation…

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Q: Suppose that f(a) = f(b) = g(a) = g(b) = 0 and the second derivatives of f and g are continuous on…

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Q: Determine the domain of the function and prove that it is continuous on its domain f(x) = cosx2 /…

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Q: Show that f is not differentiable at x = 1 and has a corner in its graph there. x 1

A:

Q: Determine the domain of the function and prove that it is continuous on its domain using Theorems…

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Q: 2. Use the limit definition of derivatives to find the expression for f'(a) for the function 1 f(x)…

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Q: Can Rolle's Theorem be applied to f(x) = 1/(1-x) on interval [0,3]? If yes, what is c in (0,3) such…

A: here we can't apply the rolles Theorem

Q: (11) (C) For the function f(x) = x° – x, determine whether the hypothesis of Rolle's theorem hold on…

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A: Consider the given function: fx=x+13-4=x+113-4 Vertical tangent at point indicates that at that…

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Q: 11. Prove from the definition that the following functions are differentiable. ()- (b) f =xy?

A: By the definition, the function f(x,y)=xy^2 is differentiable if fx(x,y) and fy(x,y) both exist.

Q: Consider the function (x)=x²+2x . Using the limit definition of the derivative, show that for any…

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Q: Suppose that f has a positive derivative for all values of x, and that f(1) = 0. Which of the…

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Q: (d) A function not continuous at 0 such that (f(x))4 is continuous at 0.

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Q: (B) Let f(x) = 3x – 1. Use the limit definition of the derivative to find f'(x). %3D

A: Explanation of the answer is as follows

Q: Show that the function g(x) =x |x | has an inflection point at (0,0) but  g " (0)  does not exist.

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A: This question is related to Real Analysis.

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A:

Q: A not uncommon calculus mistake is to believe that the product rule for derivatives says that (fg)′…

A: Derivative: f(x)=ex2We know the product rule,    ddx(u.v)=u.dvdx+v.dudx

Q: 7. Find all values of x on the graph of f(x)=(x-1)´(2x+1) at which the tangent line is horizontal.…

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Q: The limit represents the derivative of some function f at some number a. State such an f and a. V81…

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Q: ) Use the Intermediate Value Theorem to show that there is one such that f"(a) = 0. point x = a

A: As per our guidelines we are supposed to answer only first 3 subparts of the question. Kindly repost…

Q: Draw a function F(x), defined for all real numbers x, that satisfies the following properties.…

A: The first derivative is everywhere positive and the second derivative is everywhere negative