Math Problem Statement
Solution
The image presents two quadratic functions:
The goal is to find the equations of the lines that are simultaneously tangent to both curves at different points. In particular, you are tasked with finding the line that has the largest slope.
Strategy:
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Tangency Conditions: The tangents to both curves must satisfy the following:
- The slopes of the two curves at the points of tangency must be equal.
- The points of tangency must lie on both curves, i.e., where and are the points of tangency.
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Step 1: Find the slopes of the functions at any point :
- For , the derivative (slope) is .
- For , the derivative (slope) is .
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Step 2: Set the slopes equal: The slope of the tangent at point for and at point for should be equal: This gives the equation: Simplifying:
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Step 3: Set the values of the functions equal at the points of tangency: Substituting the expressions for and : Substituting : Simplifying: Bringing terms together:
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Step 4: Calculate : Using : Simplify this expression for to get the corresponding point of tangency on the second curve.
Once the points of tangency and are found, you can write the equation of the tangent line using the point-slope form.
Would you like to continue solving for and finding the final equation of the tangent line? Let me know if you'd like further explanation or any adjustments to the steps so far!
Related Questions:
- How do you determine the second tangent line with a smaller slope?
- What are the coordinates of the tangent points once you solve for and ?
- How does the slope of the tangent relate to the curvature of the parabola at different points?
- What happens to the tangency conditions if one parabola shifts horizontally or vertically?
- Can a cubic function have two tangents to two different points on a quadratic?
Tip: When working with quadratic curves, the symmetry of the parabolas can often help reduce the complexity of the calculations.
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Math Problem Analysis
Mathematical Concepts
Algebra
Calculus
Quadratic Functions
Tangency
Formulas
f'(x) = -20x
g'(x) = -2(x - 2)
f'(a) = g'(b)
f(a) = g(b)
Point-slope form: y - y1 = m(x - x1)
Theorems
Tangency Conditions
Quadratic Function Properties
Suitable Grade Level
Grades 10-12