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Suppose,x_1\; \text{and}\; x_2 are the point of maximum and the point of minimum respectively of the function f(x)=2 x^3-9 a x^2+12 a^2 x+1 respectively, then for the equality x_1^2=x_2 to be true the value of ' a ' must be

Option: 1

0


Option: 2

1


Option: 3

2


Option: 4

3


Answers (1)

best_answer

\begin{aligned} f^{\prime}(x) & =6\left(x^2-3 a x+2 a^2\right) \\ \\& =6(x-2 a)(x-a)=0 \end{aligned}

\begin{aligned} \Rightarrow \quad x & =2 a \text { or } a \\ \\f^{\prime \prime}(x) & =6(2 x-3 a) \end{aligned}

\text { If } a>0 \text {, then } x_1=a

                                           \left.\begin{array}{l} f^{\prime \prime}(2 a)=a \\ f^{\prime \prime}(a)=-a \end{array}\right] \Rightarrow x_2=2 a

\text{If }a<0, \: \text{then}\: x_1=2 a

                               x_2=a

\text{Now}, x_1^2+x_2 \Rightarrow a^2=2 a \Rightarrow a=2

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vishal kumar

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