Math Problem Statement
y=x^4-16x^2
Solution
Given the function , let's analyze it thoroughly.
1. Factor the expression:
We can simplify the expression by factoring it: Notice that is a difference of squares: Thus, the fully factored form is:
2. Find the roots:
To find the roots, set : This gives:
Thus, the roots are:
3. Determine the behavior around the roots (sign analysis):
For large positive or negative , the -term dominates, so the function grows rapidly in both directions. Now, let's analyze the sign of around the critical points .
- Interval : All terms are positive, so .
- Interval : The factor is positive, but is negative, so .
- Interval : The factor is positive, but is negative, so .
- Interval : All terms are positive, so .
4. Critical points and turning points:
To find critical points, take the derivative of .
Factor the derivative: Setting the derivative to zero: The critical points are:
5. Second derivative for concavity:
Now, take the second derivative of : Set this equal to 0 to find inflection points: Thus, the inflection points occur at:
6. Graph behavior summary:
- Roots:
- Critical points:
- Inflection points:
- The function tends towards as .
Would you like a detailed plot or further explanation on any step?
Related questions:
- How does the second derivative test determine concavity?
- How do critical points relate to maxima or minima?
- What is the significance of inflection points in graphing?
- How does sign analysis help determine intervals of positivity and negativity?
- Can you describe the symmetry of this function?
Tip: When graphing, always look at end behavior to determine how the function behaves as or .
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Math Problem Analysis
Mathematical Concepts
Algebra
Polynomial Functions
Factoring
Critical Points
Sign Analysis
Concavity
Formulas
y = x^4 - 16x^2 = x^2(x - 4)(x + 4)
First derivative: dy/dx = 4x^3 - 32x
Second derivative: d^2y/dx^2 = 12x^2 - 32
Theorems
Difference of Squares
Critical Points
Second Derivative Test
Suitable Grade Level
Grades 11-12
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