Questions: 1. Given the polynomial function f(x) = x^4 - 5x^3 - 7x^2 + 10x + 14 A. State the coordinates of the x-intercepts. B. State the coordinates of the relative maximums. C. State the coordinates of the relative minimums. D. State the range of the function using interval notation.

1. Given the polynomial function f(x) = x^4 - 5x^3 - 7x^2 + 10x + 14 A. State the coordinates of the x-intercepts. B. State the coordinates of the relative maximums. C. State the coordinates of the relative minimums. D. State the range of the function using interval notation.
Transcript text: 1. Given the polynomial function $f(x)=x^{4}-5 x^{3}-7 x^{2}+10 x+14$ A. State the coordinates of the $x$-intercepts. B. State the coordinates of the relative maximums. C. State the coordinates of the relative minimums. D. State the range of the function using interval notation. $\qquad$ $\qquad$
failed

Solution

failed
failed

State the coordinates of the \( x \)-intercepts.

Find the roots of the polynomial.

Use numerical methods to approximate the real roots of \( f(x) = x^{4} - 5x^{3} - 7x^{2} + 10x + 14 \). The real root found is approximately \( x \approx 1.54 \).

\(\boxed{x \approx 1.54}\)

State the coordinates of the relative maximums.

Find the critical points using the first derivative.

Calculate the derivative \( f'(x) = 4x^{3} - 15x^{2} - 14x + 10 \) and solve for \( f'(x) = 0 \) to find critical points.

Use the second derivative test to determine relative maxima.

Evaluate the second derivative \( f''(x) = 12x^{2} - 30x - 14 \) at the critical point \( x \approx 0.49 \). Since \( f''(0.49) < 0 \), it indicates a relative maximum. The coordinates are approximately \((0.49, 16.69)\).

\(\boxed{(0.49, 16.69)}\)

State the coordinates of the relative minimums.

Find the critical points using the first derivative.

Calculate the derivative \( f'(x) = 4x^{3} - 15x^{2} - 14x + 10 \) and solve for \( f'(x) = 0 \) to find critical points.

Use the second derivative test to determine relative minima.

Evaluate the second derivative \( f''(x) = 12x^{2} - 30x - 14 \) at the critical points \( x \approx -1.15 \) and \( x \approx 4.41 \). Since \( f''(-1.15) > 0 \) and \( f''(4.41) > 0 \), they indicate relative minima. The coordinates are approximately \((-1.15, 2.60)\) and \((4.41, -128.64)\).

\(\boxed{(-1.15, 2.60), (4.41, -128.64)}\)

State the range of the function using interval notation.

Determine the behavior of the function as \( x \) approaches \( \pm \infty \).

Since the polynomial is quartic with a positive leading coefficient, \( f(x) \) approaches \( \infty \) as \( x \) approaches \( \pm \infty \).

Evaluate the function at relative minima to find the lowest point.

The lowest relative minimum is at \( x \approx 4.41 \) with \( f(4.41) \approx -128.64 \). Therefore, the range is \([-128.64, \infty)\).

\(\boxed{[-128.64, \infty)}\)

x-intercepts: \( x \approx 1.54 \)
Relative maximums: \((0.49, 16.69)\)
Relative minimums: \((-1.15, 2.60), (4.41, -128.64)\)
Range: \([-128.64, \infty)\)

Was this solution helpful?
failed
Unhelpful
failed
Helpful

Questions asked by other users

failedAnswered
17 Multiple Answer 1 point Select all that apply. Identify the modifier(s) in the sentences provided and select the sentences that do not contain errors with the modifier(s). Don bought a Great Dane from a neighbor that was too big for the doghouse. The Great Dane that Don bought from a neighbor was too big for the doghouse. Don bought a Great Dane that was too big for the doghouse from a neighbor. 18 Multiple Answer 1 point Select all that apply. Identify the modifier(s) in the sentences provided and select the sentences that do not contain errors with the modifier(s). After an hour's search, she found the money hidden under the rug. She found, after an hour's search, the money hidden under the rug. She found the money hidden under the rug after an hour's search. 19 Multiple Answer 1 point Select all that apply. Identify the modifier(s) in the sentences provided and select the sentences that do not contain errors with the modifier(s). Marinated in mustard sauce, the cook served a delicious ham. The cook served a delicious ham marinated in mustard sauce. The cook served a delicious ham that was marinated in mustard sauce.
failedAnswered
Use the properties of logarithms to evaluate the expression. log 10^3 (Simplify your answer.)
failedAnswered
In competitive markets, firms that raise their prices are typically rewarded with larger profits.
failedAnswered
Suppose you want to have 300,000 for retirement in 25 years. Your account earns 10% interest. a) How much would you need to deposit in the account each month? b) How much interest will you earn?
failedAnswered
Donald was arrested for murdering an acquaintance. At the trial, Donald takes the stand and testifies that he and the victim knew each other. He admits that they would drink and use drugs together. One night while "partying," Donald explains in his testimony, an argument got out of hand during a card game. Donald explains that the victim accused him of cheating and pulled out a knife on him and started swinging it in the air. Donald said that he had no choice but to pull out his gun and shoot the victim. The jury convicted Donald of manslaughter. According to Marvin Wolfgang, the incident described above is best characterized as a a) misdemeanor b) criminological enterprise c) white-collar homicide d) victim-precipitated homicide