Find all the polynomials
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- 2. Find a polynomial that interpolates (-1,4) and (3,6), then find the value of f(x) at x=0 and x= 2.arrow_forwardLet P0" (x) denote the quadratic polynomial that interpolates the data {(xo, yo), (x1, yı), (x2, y2)}; let P"."(x) denote the quadratic polynomial that interpolates the data {(x1, yı), (x2, y2), (x3, Y3)}. Finally, let P3(x) de- note the cubic polynomial interpolating the data{(xo, yo), (x1, yı), (x2, y2), (x3, y3)}. Show that 14. (a) (x3 – x)P0.2) (x) + (x – xo) P!.3) (x) P3 (x) = (1,3) X3 - Xo How might this be generalized to constructing P(x), interpolating {(xo, yo), (Xn, Yn)}, from interpolation polynomials of degreen – 1? (b) ....arrow_forward(6) Let f(x) = x – 8x3. (a) Prove that f has a maximum and minimum value on [0, 4]. (b) Find the maximum and minimum values on [0, 4].arrow_forward
- 3. (a) Prove that the function f : R → R, ƒ(x) = 4x² −14x¹+30x-17, is injective. (b) Find the maximum and minimum values attained by f on the interval [—1, 1].arrow_forwardLet f(x) be a polynomial function such that f"(x) = x- 5z + 6z2. Find the r values for the point(s) of inflection for f(x). a) O r T = 2, a = 3 b) O = 0, x = 2, r ェ= -3 c) O z =-2, a =-3 3 d) O = 0, = 2, r = - 3 . a = 0, z ==2, =3 =-3arrow_forwardH2.arrow_forward
- 3) Let f(T) e R[T] be a polynomial of degree > 3. Show that there exists polynomials g(T) and h(T) in R[T], which are both not constant, i.e. of degree > 1, such that f(T) = g(T) - h(T).arrow_forwardT(x1, x2, x3) = (x1-x2, x2-x3, x1 ) is this is lineararrow_forward(1) Let f(x) = 1, x > 0 with domf= R++. Find the conjugate of function f(x).arrow_forward
- Consider the graph C in the y-plane given by the polynomial p(x), where p(x) — ах* + ba + c, b and c are real numbers, with a + 0. It is given that the point (-1,0) lies on C. Further- а, more, it is known that C has a horizontal tangent at (1, –4). Answer the following questions: and a) Set up a system of three linear equations with a, b and c as unknowns. b) Based on a), set up an augmented matrix for the linear system. c) Write the augmented matrix in b) in reduced row-echelon form. Hence, solve a, b and c. Use this information to find the equation of C.arrow_forwardHere is some information about a function ƒ : f is a polynomial function of the form f (x) = Ax² + Bx + C. f'(x) = 4x – 3 The graph off contains the point (-1,7). Use this information to determine f(x). That is, find the values of A, B , and C.arrow_forwardShow that f(x) = xª + 4x +c has at most two real roots for any number c.arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage