Question

. For the quartic function f(x) = ax4 + bx2 + cx + d, find the values of a, b, c, d such that there is a local maximum at (0, -6) and a local minimum at (1, -8). How do you find this?

Answer #1

Find a cubic function f(x) = ax3 + bx2 + cx + d that has a local
maximum value of 4 at x = −2 and a local minimum value of 0 at x =
1.

Use a system of equations to find the cubic function
f(x) = ax3 + bx2 + cx + d
that satisfies the equations. Solve the system using
matrices.
f(−1) = 10
f(1) = 8
f(2) = 34
f(3) = 94

Find a cubic function
y = ax3 +
bx2 + cx +
d
whose graph has horizontal tangents at the points (-2, 8) and
(2, 2).

The cubic polynomial P(x) = x3 + bx2 + cx + d (where b, c, d are
real numbers) has three real zeros: -1, α and -α.
(a) Find the value of b
(b) Find the value of c – d

Suppose you have a function of the general form e(x) = ax^3 +
bx^2 + cx + d
a, b, c, and d are real numbers.
Find values for the coefficients if
-there is a local maximum at (6,6)
-there is a local minimum at (10,2)
-there is an inflection point at (8,4)
Please show how you solve and explain.

The
r.v. X has the probability density function f (x) = ax + bx2 if 0
< x < 1 and zero otherwise. If E[X] = 0.6, find (a) P[X <
21] and (b) Var(X). (Answers should be in numerical values and not
be as expressions in a and b.)

. For the function f(x) = x 1+x2
(a) find the intervals on which the function is increasing or
decreasing
(b) determine the points of local maximum and local minimum
(c) find the asymptotes.

Find the absolute maximum and minimum values of f on
the set D.
f(x, y) =
4x + 6y −
x2 − y2 +
8,
D = {(x,
y) | 0 ≤ x ≤ 4, 0 ≤
y ≤ 5}
Find the absolute maximum and minimum values of f on
the set D.
f(x, y) = 2x3 + y4 +
2, D = {(x, y) | x2 +
y2 ≤ 1}

A random variable X has probability density function f(x)
defined by f(x) = cx−6 if x > 1, and f(x) = 0, otherwise.
a. Find the constant c.
b. Calculate E(X) and Var(X).
c. Now assume Z1, Z2, Z3, Z4 are independent RVs whose
distribution is identical to that of X. Compute E[(Z1 +Z2 +Z3
+Z4)/4] and Var[(Z1 +Z2 +Z3 +Z4)/4].
d. Let Y = 1/X, using the formula to find the pdf of Y.

For the function f(x)=ax3+bx2-5x+9,
determine the values of a and b so that f(-1)=12 and f'(-1)=3

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