The graph of a function f is given. Frm 3 2 Let F(x) = So f(t) dt then for 0

Q: 3. If f is continuous and f (x) dx = 8, show that the graph of ƒ intersects the line y = 4 at least…

A: The function f is continuous. The integral function is ∫13fxdx=8.

Q: Starting from the function f(x) = e¬alx| -L 1 where

A:

Q: The graph of a function f is given. -1 oV i 10 Estimate f(x) dx using five subintervals with the…

A: As per bartleby guidelines we only give first question answers.

Q: Given a function f(x) that is continuous and differentiable, and a point a in its domain, we…

A: This question can be solve by taking approximation as x tends to a (more close)

Q: Determine if the MVT applies to the function on the given interval. If so, find the points that are…

A:

Q: Find value a  such that f(x)={x^2+2x-1 if x not equal to 1…

A: given,f(x)=x2+2x-1  if  x≠1a                if x=1left hand limit at x=1limx→1- f(x)=limx→1-…

Q: 3) Let f (x) = e=x² Where is y'> 0? What does this tell you about the original function?

A:

Q: The graph of the derivative f'of a function f is shown. 2 4/ 6 8 o 12 14 16 On what interval is f…

A: Given that:   The graph of the derivatives f’ of a function is shown,

Q: 1. Suppose f is a function which satisfies f(x + y) = f(x) + f (y) for all x,y e R. What is the…

A: Let f is a function which satisfies f(x+y) = f(x) + f(y)  for all x,y∈R   f(x+y) = f(x) + f(y) ,   ∀…

Q: Find a function f (t) of the form f (t) = aet + be2t + ce' such that f(0) 1. Is it unique? = 1 and…

A: Given f(t) = ae3t+be2t+cet then f'(t) = 3ae3t+2be2t+cet As f(0) = 1, we get f(0) = ae0 + be0+ce01 =…

Q: Let A(x) = | f(t)dt, with f(x) shown in the graph below. -3 At what x values does A(x) have a local…

A: Given data is : Ax=∫0xfxdx Differentiate with respect to x, A'x=f'x

Q: Suppose f is a continuous, positive, decreasing function for x > 1 and an = f (n). By drawing a…

A:

Q: iii. Does there exist a function x e X such that 50) = f + [ e*: e x(s) ds x(1) = 1² for every t e…

A: It is given that X=C0,1 with the following norm, x=maxt∈0,1 xt The map T:X→X defined as follows:…

Q: 2) An elastic tire is described by the iven by the function f(x, y) = 4xy. Use

A: Introduction: The Lagrange methodology is employed to maximize or minimize a general performance…

Q: 81(a): Find the value of k that makes function continuous at x = 4 S f(x) = 2x + 6 x > 4 5x – k x <…

A: here f(x)=2x+6  x&gt;45x-k    x≤4according to definition of continuity,limx→-4 f(x)=limx→4…

Q: 4. Find the range of the function F(x)=√36-t² dt O [0, 36π] O [0, 18π] O [-6, 6] O [-6, 0]

A:

Q: 1. Find the discontinuity point/s of f(x) =

A: Since you have asked multiple questions in a single request we would be answering only the first…

Q: d) Find the value of the constants ' a’ and b ’ that makes the function continuous at x= 3 ах + b ;…

A:

Q: a) Let 3x2-1, for x3 f(x)= be a piecewise function. Eevaluate | Fx)dx= b) Let f(x) = 3x-1, for x< 3

A:

Q: Find the absolute maximum and absolute minimum values of f on the given interval: f(x)= square root…

A:

Q: 35. Given that f(z) is a continuous even function, then A 0 B D

A:

Q: 7) Find the absolute maximum of the function f on (0,E) S(x)= 10xe

A:

Q: The graph of a function f is given. (Let a = 2.) f 8. (a) Estimate f(x) dx using four subintervals…

A:

Q: 1) Find the value of Z that make the function continuous at x=4? V-2 x + 4 X-4 f(x) = {z3 x = 4

A:

Q: Let A(x) | f(t)dt where f is the function given by the graph below. 4 3 -1 -2 А(5) - A(9) = to 2, 3.

A:

Q: 1. Apply the Parseval's identity to the function f (x) = x, –n < x < n

A:

Q: (c) Let g : R → R be a continuous function such that +5x-9 g(t) dt = –3x . Help Willow find the…

A:

Q: For f(x) = Vx ln x, find a). domain and f', b). the intervals on which f is increasing or…

A:

Q: 4. Let g(x) = f6 S(1) dt, where f is the function whose graph is shown. (a) Evaluate g(0) and g(6).

A: g(x) = ∫0xf(t)dtHence g(x) is the area bounded by the curve f(t) and x-axis with sign.

Q: Let f(x) = ln(x² + x + 1). Find the absolute maximum and absolute minimum of f on the interval [-1,…

A: Given: f(x) = lnx2+x+1 To find: Absolute maximum and absolute minimum of f on the interval [-1, 1].

Q: The function f(x)=(1/(x+1))+4 is decreasing on what interval(s)?

A: Consider the function  fx=1x+1+4

Q: a): Find the value of k that makes function continuous at z = 4 (3x +7 z> 4 3x-k I<4 f(x) =

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

Q: -2 4. Let g(x) S) dt, where f is the function whose graph is shown. (a) Evaluate g(0) and g(6).

A:

Q: Q2/ (a):Find the points at which the function f is continuous and the points at which ƒ is not…

A:

Q: 1(a): Find the value of k that makes function continuous at x = 4 4x +8 > 4 4x – k a <4 f(x) =

A:

Q: Find the function f(x) given that f ′ (x) = e x + x and the graph of f passes through the point (0,…

A: Given: f'x=ex+x &amp; Px,y=0,3   As we know that; The Power rule: ∫xndx=xn+1n+1+C And…

Q: 7. For the function f (x) = In(x² + 1) (a) find all intervals where f is increasing, (b) find all…

A:

Q: 18. Consider the function f (x) 5e2r. On what interval(s) is f (x) %3D increasing? On what…

A:

Q: Find the value k that makes the function continuous. kx 10, x 10 M > f(x) = _ 4x + 10, x < 10 Submit…

A:

Q: ind the linear approximation to the following function at the given point a. f(x) = 4x² + x, a = 4…

A:

Q: 42) For f(x)= x-1 if 03x S 3 a) Give the domain of f b) Evaluate (-2), 0) and /(3) c) Graphf d) Is f…

A:

Q: The graph of the derivative f'(t) of f(t) is shown. Compute the total change of f(t) over the given…

A:

Q: 7. Find the c value(s) that satisfy Rolle's Theorem for the function f (x) = x* – 4x² on the…

A:

Q: On the graph of f ( x ) = sec ( x ) and the interval ( -2π, 0 ], for what value(s) of x does f ( x )…

A: f(x) = 1 on, (-2π, 0] Implies, sec(x) = 1 on (-2π, 0] i.e. x = 0 on (-2π, 0] So, x = 0  The graph of…

Q: The graph of f consists of three line segments as shown. If g (x) = So f (t) dt, find g (4) and g'…

A: Given g(x) = integral (0-x) (f(t)) dt g(4) = integral (0-4) (f(t)) dt          = F(4) - F(0)…

Q: x3-1 D. Is the function k(x) = {x²-1* * 1 continuous at x = 1? 2,x = 1

A:

Q: The graph of the derivative f'(t) of f(t) is shown. Compute the total change of (t) over the given…

A: Given: The interval given is [0,6]. Total change of f(t) over the interval [0,6] is determined as…

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.