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GivenFunction \(\text{f(x)}=5\text{x}^\frac{3}{2}3\text{x}^\frac{5}{2}, x > 0\)TheoremLet f be a differentiable real function defined on an open interval (a, b)(i) If f'(x) > 0 for all x ∈ (a, b), then f(x) is increasing on (a, b) (ii) If f'(x) < 0 for all x ∈ (a, b), then f(x) is decreasing on (a, b) Algorithm(i) Obtain the function and put it equal to f(x)Problem 19 Easy Difficulty Find the intervals on which $f$ increases and the intervals on which $f$ decreases $$f(x)=x\cos x, \quad 0 \leq x \leq 2 \pi$$ Then y= f(x) >0 from ∞ to 4, which means that y= f(x) >0 over the interval (∞, 4) Remember that y is observed in the vertical axis, and a possitive value of y imply that is y is placed above the intersection with x axis (the intersection occurs in (x,y) = (0,0) On the other hand, when y is placed below the x axis, it takes negative

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F(x) 0 over the intervals (–∞ –3) and-B) on what intervals is f'(x) Example 12 Find intervals in which the function given by f (x) = sin 3x, x, ∈ 0, 𝜋/2 is (a) increasing (b) decreasing f(𝑥) = sin 3𝑥 where 𝑥 ∈ 0 ,𝜋/2 Finding f'(x) f'(𝑥) = 𝑑(sin⁡3𝑥 )/𝑑𝑥 f'(𝑥) = cos 3𝑥 × 3 f'(𝒙) = 3 cos 3𝒙 Putting f'(𝒙) = 0 3 cos 3𝑥 = 0 cos 3𝑥 = 0 We know that cos θ = 0

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A helpful shorthand f ′ > 0 f ↑Let f (x) = x25(1−x)75,xϵ0,1 ⇒ f ′(x) =25x24(1−x)75 −75x25(1−x)74 = 25x24(1−x)74{(1−x)−3x} = 25x24(1−x)74(1−4x) We can see that f ′(x) is positive for x< 41 and f ′(x) is negative for x> 41This captures an intuitive property of continuous functions over the real numbers given f continuous on 1, 2 with the known values f(1) = 3 and f(2) = 5, then the graph of y = f(x) must pass through the horizontal line y = 4 while x moves from 1 to 2 It represents the idea that the graph of a continuous function on a closed interval can be drawn without lifting a pencil from the

 Find The Intervals In Which The Function F Given By F X Sinx Cosx 0 X 2π Sarthaks Econnect The function f(x) = tan^ 1(sinx cosx), x > 0 is always an increasing function on the interval asked dec 22, 19 in limit, continuity and differentiability by rozy ( 418k points) applications of derivativesSolution for Find the intervals where f"(x) < 0 or f"(x) > 0 as indicated f"(x) < 0 (1, 0) O (1, 0) (00, 1) (1, 0), (1, o0) Answered Find the intervals where f"(x) < 0 or bartleby menu1 (this gets some marks) 8 < f(0) = 0 2e0 = 21 = 1

Solution for Given the graph 13 On what intervals of x is f(x) increasing?For x = 0 (1 6 k ) lo g (3 1 0 ) lo g 2;For x = 0 is continous at x = 0, then the value of k is

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 Using the Key Idea 3, we first find the critical values of f We have f ′ (x) = 3x2 2x − 1 = (3x − 1)(x 1), so f ′ (x) = 0 when x = − 1 and when x = 1 / 3 f ′ is never undefined Since an interval was not specified for us to consider, we consider the entire domain of f which is (− ∞, ∞)4x 2 ex = 0 Consider the function f(x) = 4x 2 ex Equivalently, we must show that f(x) = 0 for some x in the interval (0;More_vert Where on the interval 0, 4 does f ( x ) = 4 x – x 2 have a horizontal tangent line?

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Let $A(x)=\int_{0}^{x} f(t) d t$, with $f(x)$ as in Figure 12 Determine (a) The intervals on which $A(x)$ is increasing and decreasing (b) The values $x$ where $A(x)$ has a local min or max (c) The inflection points of $A(x)$ (d) The intervals where $A(x)$ is concave up or concave downIf f (x) = ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ 1 − cos 8 x 2 0 x 3 x − 6 x − 1 0 x ;M−∞) such that f(x) ≤M(or f(x) ≥m, resp) for all x∈I So, fis bounded on Iif it is bounded both above and below on I We will be most interested in Ibeing an open or a closed interval Examples (1) Function f(x) = x2 is bounded on I= (−1,1) Indeed, we have 0 ≤f(x)

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On the interval (1, 3), the function f(x) = 3x 2/x is (A) Strictly decreasing (B) Strictly increasing asked in Limit, continuity and differentiability by Rozy (Use n = 4 Solution First, divide the interval 0, 2 into n equal subintervals Using n = 4, Δx = (2 − 0) 4 = 05 This is the width of each rectangleDefinition of an increasing function A function f(x) is "increasing" at a point x 0 if and only if there exists some interval I containing x 0 such that f(x 0) > f(x) for all x in I to the left of x 0 and f(x 0) < f(x) for all x in I to the right of x 0

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Click here👆to get an answer to your question ️ The function f(x) = x^x decreases on the interval Join / Login Question The function f (x) = x x decreases on the interval A (0, e) B (0, 1) C 1 lo g x < 0 lo g x 0 for all x on (a,c) and f'(x)0 for all x on (c,b), then f(c) is a local maximum value 3) If f'(x) has the same sign on both sides of c, then f(cMath 151 cBenjamin Aurispa 51DerivativesandGraphs What does f′ say about f?

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Incoming Term: f(x) 0 over the intervals (–∞ –3) and, f(x) 0 over the intervals (–∞ –3) and brainly,