(3a) Show that the formula in the Mean-Value Theorem can be written as follows: f(x+h)-f(x) h = f'(x+0h) where 0 € (0, 1). (b) Determine as a function of r and h when f(x) = 1².

Q: 3)Verify Rolle's Theorem for the function f (x) = 2x² + 3 defined over the interval -5, 5]. Justify…

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Q: If  f′(x) < 0 for 1 <x< 6, then f is decrescent over(1,6) True or false and why

A: If  f′(x) &lt; 0 for 1 &lt;x&lt; 6, then f is decrescent over (1,6) True or false and why

Q: Find all values of x for which the tangent line of f(x) x2 + 18x + 96 is horizontal.

A: fx=x2+18x+96 Differentiating with respect to x, we get f'x=ddxx2+18x+96=12x2+18x+96·dx2+18x+96dx…

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Q: 3)Verify Rolle's Theorem for the function f(x) = 5² + 3 defined over the interval [-4, 4]. Justify…

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Q: . Solve the following a) Simplify f(x+h) - f(x) where f(x) = x^2-2x b) If f(x) = (x-2) (x^2-1),…

A: f(x)=x2-2x Now we find out f(x+h). Replace x with x+h f(x)=x2-2xf(x+h)=(x+h)2-2(x+h)=x2+2xh+h2-2x-2h…

Q: 6. Verify Mean Value Theorem for the function f(x) = 3x² -4x +5 in [-1,1].

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Q: find fz (2²+1) da can not be done 0 x¹ + x² + c © x³ + x +1 Onone of these answers

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Q: 17. Sketch the graph of f(x) = { 2x + 2, x ≤ −1 [x³−x, x>−1 and find the derivative. Determine…

A: 17. Sketch the graph of  and find the derivative.Determine whether  satisfies the conditions of the…

Q: Show that Rolle’s theorem is false for the function f(x) = |x| on the interval [−2, 2]. How can a…

A: Given function f(x)=|x|interval [-2,2] We apply Rolle's theorem only when the function is continuous…