Exams Joint Entrance Examination Mathematics Application of Derivatives Question 56 Question 56 of 212 Medium NUMERICAL
If the set of all values of a a a , for which the equation 5 x 3 − 15 x − a = 0 5 x^3-15 x-a=0 5 x 3 − 15 x − a = 0 has three distinct real roots, is the interval ( α , β ) (\alpha, \beta) ( α , β ) , then β − 2 α \beta-2 \alpha β − 2 α is equal to \_\_\_\_\_\_\_\_\_.
Answer: 30
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5 x 3 − 15 x − a = 0 f ( x ) = 5 x 3 − 15 x f ′ ( x ) = 15 x 2 − 15 = 15 ( x − 1 ) ( x + 1 ) \begin{gathered} 5{x}^{3}-15x-a=0 \\ f(x)=5{x}^{3}-15x \\ {f}^{'}(x)=15{x}^{2}-15=15(x-1)(x+1) \end{gathered} 5 x 3 − 15 x − a = 0 f ( x ) = 5 x 3 − 15 x f ′ ( x ) = 15 x 2 − 15 = 15 ( x − 1 ) ( x + 1 )
a ∈ ( − 10 , 10 ) α = − 10 , β = 10 β − 2 α = 10 + 20 = 30 \begin{gathered} a\in (-10,10) \\ \alpha =-10,\beta =10 \\ \beta -2\alpha =10+20=30 \end{gathered} a ∈ ( − 10 , 10 ) α = − 10 , β = 10 β − 2 α = 10 + 20 = 30 Answer the question to unlock