Given the function f(x) = x^4 + x^3 - 35x^2 - 57x + 90, answer the following questions: a) How many extreme points does it have? b) What is the degree? c) Is the degree even or odd... Given the function f(x) = x^4 + x^3 - 35x^2 - 57x + 90, answer the following questions: a) How many extreme points does it have? b) What is the degree? c) Is the degree even or odd? d) Identify the sign of the leading coefficient. e) X-intercepts: f) Based on the degree how many roots does it have (roots could be real, imaginary, or they can repeat)? g) What is another name for x-intercepts? h) Y-intercept: i) End behavior: j) Find the intervals in which the graph is increasing: k) Find the intervals in which the graph is decreasing: l) Find the average rate of change from x=-1 to x=3. m) Find an interval with average rate of change of 400.
Understand the Problem
The question provides a polynomial function and asks multiple related questions about its properties, including extreme points, degree, leading coefficient, intercepts, end behavior, and intervals of increase and decrease. The overall aim is to analyze the characteristics of the polynomial function.
Answer
a) 3, b) 4, c) Even, d) Positive, e) Solve $f(x) = 0$, f) Up to 4, g) Roots, h) (0, 90), i) $\infty$, j) Analyze $f'(x)$, k) Analyze $f'(x)$, l) 16.5, m) Find where average = 400.
Answer for screen readers
a) 3
b) 4
c) Even
d) Positive
e) Solve $f(x) = 0$ for x-intercepts.
f) Up to 4 roots.
g) Roots or zeroes.
h) (0, 90)
i) Both end behaviors tend to $\infty$.
j) Analyze $f'(x)$ for increases.
k) Analyze $f'(x)$ for decreases.
l) Average rate of change from $x=-1$ to $x=3 = 16.5$.
m) Find where the average rate of change = 400.
Steps to Solve
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Identify the degree of the polynomial The given function is $f(x) = x^4 + x^3 - 35x^2 - 57x + 90$. The degree is determined by the highest power of $x$, which is 4.
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Determine whether the degree is even or odd Since 4 is an even number, the degree of the polynomial is even.
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Identify the leading coefficient The leading term is $x^4$, and its coefficient is 1, which is positive.
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Count extreme points The maximum number of extreme points for a polynomial of degree $n$ is $n-1$. Thus, for degree 4, the maximum number of extreme points is $4 - 1 = 3$.
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Find the x-intercepts To find the x-intercepts, set $f(x) = 0$. This requires solving the polynomial equation, typically needing numerical methods or graphing for exact values.
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Determine the number of roots A polynomial of degree 4 can have up to 4 roots (real, imaginary, or repeated).
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Another name for x-intercepts X-intercepts are also called "roots" or "zeroes" of the function.
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Calculate the y-intercept To find the y-intercept, evaluate $f(0)$: $$f(0) = 0^4 + 0^3 - 35(0^2) - 57(0) + 90 = 90$$. Thus, the y-intercept is (0, 90).
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Analyze the end behavior For large positive ($x \to \infty$) or negative ($x \to -\infty$) values of $x$, the leading term $x^4$ dominates:
- As $x \to \infty$, $f(x) \to \infty$
- As $x \to -\infty$, $f(x) \to \infty$
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Find intervals of increasing and decreasing To find these intervals, compute the derivative $f'(x)$ and analyze its critical points: $$ f'(x) = 4x^3 + 3x^2 - 70x - 57 $$ Set $f'(x) = 0$ and solve for $x$.
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Calculate average rate of change from x=-1 to x=3 The formula for average rate of change is: $$ \text{Average Rate} = \frac{f(3) - f(-1)}{3 - (-1)} $$ Calculate $f(3)$ and $f(-1)$ then substitute back into the formula.
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Find an interval with average rate of change of 400 This involves finding intervals where the average rate of change (slope) equals 400. Analyze the derivative or use numerical methods to find such an interval.
a) 3
b) 4
c) Even
d) Positive
e) Solve $f(x) = 0$ for x-intercepts.
f) Up to 4 roots.
g) Roots or zeroes.
h) (0, 90)
i) Both end behaviors tend to $\infty$.
j) Analyze $f'(x)$ for increases.
k) Analyze $f'(x)$ for decreases.
l) Average rate of change from $x=-1$ to $x=3 = 16.5$.
m) Find where the average rate of change = 400.
More Information
The function's degree of 4 indicates that it is a quartic polynomial. Quartic polynomials can have various shapes and behavior due to their number of extreme points. The derivatives help in determining increasing and decreasing intervals, providing insight into the shape and direction of the graph.
Tips
- Forgetting to check both ends of the polynomial when analyzing end behavior.
- Confusing x-intercepts with y-intercepts.
- Miscalculating the average rate of change by not using the correct formula.
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