Consider the function f(x) = -x² + 4x – 2. a) Sketch the graph of f in the interval [-1, 5]. (6 marks) b) Find the equation of the tangent line to the graph of f at r = 3, and show it on the graph of f. (4 marks)

Q: Use the first and second derivatives of (x)= x - 3x + 1 to determine the intervals on which f is…

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Q: Find the interval of x-values on which the curve is concave up and concave down , then sketch the…

A: Since you have asked multiple question, we will solve the first three subparts for you. If you want…

Q: Two tan gents are drawn to the function f(x) = x2 - 4x %3D + 10 through the point (1, -9). Find the…

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Q: 2. Graph the function f (x)= x+ Vlx\ . Zoom in repeatedly, first toward the point (-1, 0) and then…

A: Note: There are multiple questions. No specific question has been asked.So, We will solve the first…

Q: Use the second order derivative test to find local exterma of the function  f(x) =x3-9x2-48x+72 on…

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Q: Show that no tangent line to the graph of f(x) = tan x has zero slope. What is the least slope of a…

A: Slope of tangent line to f(x)=tanx is given by f'(x)=sec2(x). sec(x)≠0 , so f'(x)≠0  This proves…

Q: By considering the derivative of the function f: [-1,1] → R defined by f(r) = show r2 +1' that f-1…

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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: The function f(x) is defined by f(x) =2. What points (x, y) on the graph of f(x) have the property…

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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: Consider the function f(x)f(x) whose second derivative is f′′(x)=9x+10sin(x). If f(0)=3f(0)=3 and…

A: We have to find f(x) and find f(3) using given intial condition

Q: Find the intervals of monotonicity of the function f(x)=x+sinx.

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Q: 2. Find the derivative and the analytic domain of f(2) = 1+ %3D 22+1

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Q: Let f(x) = x- 5x2. Find the point(s) on the graph of f where the tangent line is horizontal. (х, у)…

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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: Use the definition of a derivative to determine all points for the function f(x) = ((x+|x|)^2)+1 is…

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Q: Sketch the graph of the function f(x) = 12 x sin x on the interval [−3π, 3π]

A: we have to sketch the graph of the given function at the given interval. f(x)=12xsinx on the…

Q: Determine the open intervals on which the graph of the function is concave upward or concave…

A: y=x+2sin x Let take first derivative y=x+2cscxy'=1-2cotxcscxNow take second…

Q: Let f(x) be a function that is differentiable at x = -2. Given that f'(-2) = 3 and f(-2) = 6, find…

A: Given data: f'(-2) = 3 x1 = -2 f(-2) = y1 y1 = 6

Q: Find the derivative h(x) = h'(x) = of the transcendental function. 14 sec x

A: The given function hx=5x-14sec x we have to find the derivative of h(x)

Q: calculate the are between the curves f(x)=x^2 and g(x)=x^3 on the interval [-1,1]

A: Given,

Q: 4) Find all values in the interval [-27,27]at which the graph of f has a horizontal tangent line.…

A: topic- functions and derivatives

Q: х — 7 - f(x) = %3D 5х + 7 Find an equation for the tangent line to the graph of f at x = 3.

A: Given: fx=x-75x+7 To find the tangent line, we find the slope Therefore differentiate with respect…

Q: 10. Find a function f such that f'(x) =r' and the line 4x - ļy = 3 is tangent to the graph of f. %3D

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Q: Draw the y = sinx and y = 1- x² on the domain -2 <x<2 and filling the between the funtions.

A: the filling part is given below-

Q: 1.Use the second sufficient condition for an inflection point (the third derivative test) to find…

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Q: 2. Consider the function f(x) = 1 – x², z ER. Draw the graph of f and calculate the equation of line…

A: solve the following

Q: Consider the function f(x) = 1 - 7x² OI function on this interval, i.e. f(3)-f(-4) 2 4)

A: answer is in next step . please give a like !!

Q: Find a function f such that f'(x) = x4 and the line 2x - y = 3 is tangent to the graph of f.

A: Given that

Q: Find the x-values at which f is differentiable. (Enter your answer using interval notation.)

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Q: 20 Let f(x) = 4x + x2 given by y = mx + b for . Then the equation of the tangent line to the graph…

A: We have to find

Q: Suppose that a one-to-one function f has tangent line y=5x+3 at the point . (1,8)Evaluate (f-1)'8

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Q: Use the definition of the derivative (the limit of a difference quotient} to find a formula in x for…

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Q: Let f(x) = 33 –x The slope of the tangent line to the graph of f (=) at the point (-5, 8) is The…

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Q: Find the x-value for all relative extrema of f(x) using the first derivative test, and specify…

A: f(x)=x3/3 + x2/2 -2x Find the x-value for all relative extrema of f(x) using the first derivative…

Q: Consider a function f continuous and differentiable on R with f(2. 5) = -3. 1 and f'(2. 5)=-0. 2.…

A: We have to find

Q: Let f(x) = x³ – 2x2. Find the point(s) on the graph of f where the tangent line is horizontal. (x,…

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Q: he tangent line to y= f(x) at (-7, -6) passes through the point (-9, -2). Compute the following.  a)…

A: GIVEN DATA : Equation y=fx                         Tangent of the y=fx at -7,-6 passes through the…

Q: 3. Consider the function 8x f(x) = + 15. - The tangent line to the graph of f at the point (6, f(6))…

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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: Make a sign diagram for the derivative of the rational function. 6 f(x) = x + 3 f'(x) < 0 undefined…

A: Given function, fx=6x+3

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) be defined by f(x) = 3x² + 5x – 40. Let (a, b) be the point on the curve y = f(x) such that…

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Q: 9. Given the function f(x) 3x*-4x2 +2, use the first and second derivative tests to find: a) the…

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Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.  

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.