Question

Find a such that f(x) will be continous everywhere

f(x)= ( x+1) 1<x<3

x^2+bx+c x< or equal to 1 and x > or equal to 3

Answer #1

Find the values of a and b that make
f continuous everywhere.
f(x) =
x2 − 4
x − 2
if x < 2
ax2 − bx + 3
if 2 ≤ x < 3
2x − a + b
if x ≥ 3

a)
Find f(x) is f(x) is differentiable everywhere and
f'(x)= { 2x+8, x<2
3x2, x>2
given f(1)=1
b)
the point (-1,2) is on the graph of
y2-x2+2x=5. Approximate the value of y when
x=1.1. Then use dy/dx and
d2y/dx2 to determine if the point (1,-2) is a
max, min, or neither.

1. Let f(x)=x^2−4x+3/x^2+2X-3 Calculate lim x→1 f(x)
by first finding a continuous function which is equal to ff
everywhere except x=1 .
2. Use the chain rule to find the derivative of the following
function 5(−2x^3−9x^8)^12

Let f(x) be a cubic polynomial of the form x^3 +ax^2 +bx+c with
real coefficients.
1. Deduce that either f(x) factors in R[x] as the product of
three degree-one
polynomials, or f(x) factors in R[x] as the product of a
degree-one
polynomial and an irreducible degree-two polynomial.
2.Deduce that either f(x) has three real roots (counting
multiplicities) or
f(x) has one real root and two non-real (complex) roots that are
complex
conjugates of each other.

Use a system of linear equations to find the quadratic
function
f(x) = ax2 + bx + c
that satisfies the given conditions. Solve the system using
matrices.
f(1) = 3, f(2) = 12, f(3) = 25
f(x) =

The function f(x) = x^3+ax^2+bx+7 has a relative extrema at x =
1 and x = -3.
a.) What are the values of a and b?
b.) Use the second derivative test to classify each extremum as
a relative maximum or a relative minimum.
c.) Determine the relative extrema.

1) find a cubic polynomial with only one root
f(x)=ax^3+bx^2+cx +d such that it had a two cycle using Newton’s
method where N(0)=2 and N(2)=0
2) the function G(x)=x^2+k for k>0 must ha e a two cycle
for Newton’s method (why)? Find the two cycle

Prove that if f(x) is continous on [a,b], then f(x) is integrable
on [a,b].

find the values of a,b and c for the quadratic polynomial f(x)=
ax2+bx+c which best fits the function h(x)=w^2x+1 at
x=0
f(0)=h(0), and f'(0)=h'(0), and f''(0)=h''(0)
check your answer by graphing both f and h on www,desmos.com and
zoom in around x=0. Explain what you notice.

About convex function:
(1) Please show that f(x) = ax2 + bx + c is a convex
function if and only if a ≥ 0.
(2) Please show that f(x) = 1/2·xTQx+aTx
is a convex function if and only if Q is positive
semi-definite.

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