1 f (x) = 1 if x is irrational if x is rational

Q: The function - 7x – 10 - f(x) x – 3 has a single critical number at x = A. Find this critical…

A: The critical number of a function is obtained by solving f'(x)=0 for x. The quotient rule of…

Q: 1) Find the absolute extremum values of f(x) = x² - 3x + 12 over the interval 1, 3]. -

A: The extreme value of function at f(3)= 30

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: 1. Explain why both f(x)= x³+ 1 and g(x) tions. x3+1 are rational func-

A:

Q: 7. Find all the critical points of f(x) = (x² – 4x – 4) e² %3D and classify them.

A:

Q: f f(x) = ln(x2+10x+16) ov

A: The absolute minimum of f(x) is 6ln(2) and the absolute maximum of f(x) is 4ln(2)

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

A:

Q: snip

A: Consider the rational function R(x)=x-1x2-4. Recall that the vertical asymptotes of a ration…

Q: If R(x) = p(x)/q(x) is a rational function and if p and q have no common factors, then R is ____ .

A: If R(x) = p(x)/q(x) is a rational function and if p and q have no common factors, then R is ____ .

Q: 1) Find the absolute extremum values of f(x) = x² – 4x + 16 over the interval [1, 4].

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Q: x*y"-xy'+y = 0 Y, = In x

A:

Q: 4-The rational function y=ax?/(bx2+(2b+1)x +b) has one vertical asymptote and passes through point…

A: Solution

Q: 5. Find the domain of the rational function f(x) = 3x 2 - 8x - 3 2x 2 + 7x- 4. D a. D(-co, -4) U…

A:

Q: Prove that for integers y=1 the equation 3h-n (2x₁ + x₂ + 4x₂) +3² (~1(2x₁+x₂+.+xn-1)+...+3^-1 (2²₁)…

A: We will use contradiction to prove it.

Q: 1. Solve the lowest term of the following rational expres= functions. Show your complete solutions.…

A:

Q: on 15 e quadratic approximation of the function f (x) =5x³– 4x+ 12 at x=1 is: P 15x2 – 19x+ 17 P…

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Q: The function f(x)=2x^3−36x^2+162x−9 has two critical numbers. The smaller one is x = and the…

A:

Q: Consider the rational function 2x5 f(x) = 9x4 + 4x3 – 18x² – 28x + 12 2x3 x2 + 6x – 3 where 2x – x²…

A: Check the degrees of the numerator and denominator to determine the partial fractions. Now…

Q: 7. Find the critical numbers of the function. (1) ƒ (x)= x³ + r² – r

A:

Q: -1 is a critical number of the function f(x) = V2x + x².

A: derivative

Q: x*y"-xy'+y = 0 c) Y, = In x

A: Let's find.

Q: 1) Find the absolute extremum values of f(x) = x² – 3x + 12 over the interval [1, 3].

A:

Q: 3. Use Rolle's Theorem to show that the function f (x) = x² sin(x) – x sin(x) has a critical point…

A: For y = f(x). Let f(x) is continuous at every point of the closed interval [a, b] and differentiable…

Q: Express the rational functions as partial fractions: x² – 4

A:

Q: Express the rational functions as partial fractions: 1 (x - 1)(x² + 1)

A: We need to express the rational functions as partial fractions.            1x-1x2+1

Q: Consider the rational function f(x)=8x/x−5. a)As x→−∞, f(x)→ b)As x→∞, f(x)→

A: We have to find limit

Q: Let f(x)=(7+3x)4   f(x) has one critical value at A =

A:

Q: Write the rational function as a sum of partial fractions: x+1 A B -- A = J.B = + (x-4)(x+7)¯ x-4…

A: Given that, x+1x-4x+7=Ax-4+Bx+7  ----(1) To find A and B, Taking least common divisor on the right…

Q: sketch the following rational functions by determining all critical values. h(x)=x2-2x-8/x2-9

A: To evaluate the critical points of the rational function h(x) = 0 we must solve the equation h'(x) =…

Q: 1) Find the absolute extremum values of f(x) = x² – 3x + 12 over the interval [1,3]. [2 marks] %3D -

A: Given: fx=x2-3x+12 over the interval 1,3

Q: 6. Determine f such that f"(x) = 12x – 1 and 1 is both a critical number and a y- intercept of f.

A: A differential equation is an equation in x,y and derivatives of y. A linear differential equation…

Q: 4. Find the x-intercepts of the rational function y = -32. x² –x-2 Ох%3D-2,1. Од%3D 2,—1. O x = 2,…

A:

Q: 1) Find the absolute extremum values of f(x) = x² – 3x + 15 over the interval [1,3].

A: To find the extreme values of the given function f(x) = x2 - 3x + 15 over [1, 3]

Q: The function f(x) = 2x° – 36x2 + 162x – 11 has two critical numbers. The smaller one equals and The…

A:

Q: What is the critical point of the function g (x) = (x ^ 2/2) + x (ln (x) −1), using Newton-Raphson,…

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Q: Is f(x) =x−3/x + 1 a rational equation? Prove the answer with a solution if possible.

A:

Q: Find the critical numbers of the functions. (3) f(x)= |x-1| 1 (4) f(x)= (x² –x)

A: Critical numbers are those points on the function where its derivative becomes zero or infinite.

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: 1) Find the absolute extremum values of f(x) = x2 – 4x + 18 over the interval [1, 4]. [2 marks]

A:

Q: 1) Find the absolute extremum values of f(x) = x² – 3x + 10 over the interval [1, 3].

A:

Q: x = -1 is a critical number for g(x) = xª +x³ +x - 1. But the second derivative test fails for x =…

A: Explanation of the answer is as follows

Q: 3/ Find the critical numbers of the function f(x)= x/5 (4-x)

A: The given function f(x)=x354-x

Q: Simplification of In x + In(x + 1) = 0 is Ox2 + x-1= 0; x > 0 Ox2 +x +1%= 0; x > 0 %3D x2 +x-e = 0;…

A: Given: ln(x)+ln(x+1)=0

Q: 8) State the critical points of the function k(x) = x2 In x2.

A: Given,          The function kx=x2lnx2

Q: Find the critical numbers of the function. f(x)=32x-2x^2 x=

A:

Q: 1. If the rational function y = r(x) has the vertical asymptote %3D x = 2, then as xr→2*, either…

A: Given: y=rx

Q: 1. f(x) = x2 + x – 1 for all values of x %3D

A: A detailed solution is given below

Q: Find the domain of the rational function. X-7 f (x) = x+9 Enter your answer in interval notation. To…

A:

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.