Find a positive real number a such that the tangent lines to the graphs of two functions f(x) = x3 – x2 - 23x+ 19 and 2x3 g(x) = x² +26x+17 at x = a are parallel. 3

Q: 1) Find the absolute extremum values of f(x) = x² – 3x + 20 over the interval [1, 3]. [2 - marks]

A: We have to find the absolute extremum values of the given function.

Q: 7. Use calculus to find the exact values of the x-coordinates of all points of the graph of f(x) =…

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Q: Prove, without graphing, that the graph of the function has at least two x-intercepts in the…

A: Given   y = sin (x3) ....(1)   , (1, 2)

Q: For the function, f(x) = 9x² – x³ - a. Use the derivative to find all critical points b. Classify…

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Q: Find the local maximum and local minimum values of f using both the first and Second derivative…

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Q: 4. Find the values of x for which the line tangent to the graph of f(x) = 3x2 – 4x + 8 has a tangent…

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Q: Find the absolute extrema of the function on the closed interval.                   f(x) = x3 + 6x2…

A: To find the absolute extrema of the function on the closed interval. first differentiate and find…

Q: Let f(x)=(x-2)(x2+4x-7). Find all the points on this curve where the tangent line is horizontal.

A: The given function is f(x)=(x-2)(x2+4x-7). Since, the slope if the horizontal tangent line is zero.…

Q: Prove, without graphing, that the graph of the function has at least two x-intercepts in the…

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Q: Let f(x)=x+(1−x)^(1/2) Find the local maximum and minimum values of f using both the first and…

A: For local maxima or minima:The first derivative test is f'(x) = 0The second derivative test…

Q: Find the absolute maximum and absolute minimum values of f on the given interval. f(x) = ln(x2 + 5x…

A: The given function f(x) = ln(x2 + 5x + 13),    [−3, 1] Differentiate with respect to x…

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: 4. Find the values of x for which the line tangent to the graph of f(x) = 4x2 – 8x + 4 has a tangent…

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Q: Find the local maximum and minimum values of f using both the First and Second Derivative Tests. x2…

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Q: Find the equation of the tangent line to the graph of the functic f(x) = 3x² - 4x +5 at the point…

A: Find out the equation of tangent line.

Q: The function f(x) = 2x3 – 39x? + 240x + 9 has derivative f'(x) = 6x? – 78x + 240. %3D f(x) has one…

A: Given data:   The expression for the given function is f(x)=2x3-39x2+240x+9. The expression for the…

Q: Find the absolute extrema of f(x) = x3 +5x2-8x+2 on the interval [−1, 2].

A: Given function is fx=x3+5x2-8x+2 ; x∈-1, 2

Q: Find all critical points of the function f(x) = 2 – 2x + 2. %3D

A: topic - derivatives

Q: Let f(x) = x³ + 3x2 + x + 3. Determine the points f where the tangent line is horizontal on

A: to find the points where the tangent line is horizontal we have to first derivate the function and…

Q: Find and classify the stationary points of f(x) 2x} + 4x,x3 – 10x,x, + x}

A: Given,     f(x)=2x13+4x1x22-10x1x2+x22

Q: Find the constant a such that the function is continuous on the entire real line. 4sin x if x < 0…

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Q: Find all critical points of the function f(x) = x² – 4x + 6 = x

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Q: Find the critical points of the function f(x) = 8x^3 + 4x^2 - 12x + 10 and determine their nature.

A: Solution: Given: f(x)=8x3+4x2-12x+10

Q: Find the Quadratic approximation of the function at the given value f(x) = 1n(x + 1)     near a = 0

A: Given  The function is f(x) = ln(x + 1)     near a = 0 .   Formula for Quadratic approximation :-…

Q: Use Intermediate Value Theorem and Bolzano's Theorem to show that f(x)=x^3 and g(x)=2x+3 intersect…

A: given functions are                                                     f(x)=x3g(x)=2x+3 if both the…

Q: Find the absolute maximum value of f(x) = x2 – 2x + 10 over the interval 0, 3 -

A: Find your answer below

Q: Find the critical numbers of the function.…

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Q: 3. Find all -values where the graph of f(x) = x³ + 3x² + x + 3 has a horizontal tangent line.

A: Solution is given below:

Q: Locate the critical points of the function f(x) = x + 2x + 6x - 4 and use the second derivative test…

A: We solve the equation f'(x)=0 to find the critical points of the function. If a is a critical point,…

Q: Find a point c satisfying the conclusion of the MVT for the function y(x) = 4x/3 on the interval [0,…

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Q: Find the x-coordinates of all points where the graph of f(x) = 2x^3+9x^2-168x+21 has a horizontal…

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Q: In the function:  f(x)= (3x^2)ln(x) , x&gt;0 What are the x-coordinates of all local minima and…

A: Given:f(x)= (3x^2)ln(x) , x&gt;0

Q: Find the maximum value of x if f(x)=5 sin x – Using Quadratic Interpolation with a relative error of…

A: Given: If f(x)=5sin x-x610, find the largest value of x. Using Quadratic Interpolation with a 1%…

Q: Show that if the graph of a twice-differentiable function ƒ(x) has an inflection point at x = a,…

A: Find the reason why the tangents lines fit well at the inflation points

Q: Find all points on the graph of F(x)=-x3+3x2-2 at which there is a horizontal tangent line.

A: Given curve is Firstly, we will find derivative of F(x) We now that slope of tangent to a curve…

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

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Q: Let f(x)=x^3-4x^2. Find the points on the graph of f where the tangent line is horizontal.

A: f(x) = x3 - 4x2Horizontal tangent when f'(x) = 0differentiate f(x) with respect to 'x'f'(x) = 3x2 -…

Q: for the funciton f(x) = x^3 - 75x Its local maximum  is____, which occurs at _____> Its local…

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Q: The function f(x)=2x^3−42x^2+270x+8has derivative f′(x)=6x^2-84x+270. f(x) has one local minimum…

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Q: Find all critical points for the function f(x)= x²e!-2¤. %3D

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Q: 5. Use the second derivative test to determine whether f(x) has local maximum or local minimum at…

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Q: 1) Find the absolute extremum values of f (x) = x² – 4x + 9 over the interval 1,4| -

A: Given,  fx = x2-4x+9    We have to find the absolute extremum values in the interval 1, 4

Q: The distance between the x-intercept of the tangent line to the curve y = and the x-coordinate of…

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Q: Find the local maximum and minimum values of f using both the First and Second Derivative Tests.…

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Q: 46. Let f(x) = x³ - 4x² . Find the points on the graph of f where the tangent %3D line is…

A: 46. The given function is fx=x3-4x2. Differentiate fx with respect to x. f'x=ddxx3-4x2=3x2-8x Now,…

Q: Find the maximum value of the function (x1 + 4.x2)² f(#1, x2) = by expressing f (x1, x2) as the…

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Q: 5. Use the second derivative test to determine whether f(x) has local maximum or local minimum at…

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Q: How did the function f(x) = -x^3 + e^x - 8 satisfy the intermediate value theorem? Show that f has a…

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Q: The degree of f(x) is v and the leading coefficient is There are distinct real zeros and relative…

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Q: Find the critical points of f(x)=3x^4 -4x^3 +5 and use the first derivative test to determine local…

A: Given,          fx=3x4-4x3+5

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: