Let αθ and βθ be the distinct roots of 2x2+(cosθ)x−1=0,θ∈(0,2π). If m and M are the minimum and the maximum values of αθ4+βθ4, then 16(M + m) equals:
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Step-by-Step Solution
Step 1: Find sum and product of roots
For a quadratic equation ax2+bx+c=0, the sum of the roots is -b/a and the product of the roots is c/a. Here, a=2, b=cosθ, and c=−1. We use these formulas to find the sum and product of αθ and βθ.
Step 2: Express sum of fourth powers in terms of sum and product of roots
We want to find the value of αθ4+βθ4. We can express this in terms of (αθ2+βθ2) and (αθβθ). We also know that αθ2+βθ2=(αθ+βθ)2−2αθβθ.
Step 3: Substitute values and simplify
Now we substitute the expressions for (αθ+βθ) and (αθβθ) into the formula from the previous step. We then simplify the expression to get a function of cosθ.
Step 4: Further simplification
Continuing the simplification, we expand the terms and combine them. Let y=cosθ. Since θ∈(0,2π), the range of cosθ is [−1,1]. The problem states that the roots are distinct, which means the discriminant must be positive. The discriminant is b2−4ac=(cosθ)2−4(2)(−1)=cos2θ+8. Since cos2θ≥0, the discriminant is always positive, so the roots are always distinct for any θ.
Step 5: Find minimum and maximum values
Let f(y)=161y4+21y2+81, where y=cosθ. Since y2 is involved, we can let z=y2. Then z∈[0,1]. The function becomes g(z)=161z2+21z+81. This is a parabola opening upwards, so its minimum and maximum values on [0,1] occur at the endpoints. For z=0, g(0)=81. For z=1, g(1)=161+21+81=161+8+2=1611. Thus, m=81 and M=1611.
Step 6: Calculate 16(M+m)
Finally, we substitute the values of M and m into the expression 16(M+m) and calculate the result.
Step 7: Final Calculation
We add the fractions inside the parenthesis and then multiply by 16 to get the final answer.