Maths · Graphs of simple functions
Let =x^{3}-12 x \) be function such that the equation |= \) \) has exactly disti
Let \( f(x)=x^{3}-12 x \) be function such that the equation \( |\boldsymbol{f}(|\boldsymbol{x}|)|= \) \( \boldsymbol{n}(\boldsymbol{n} \in \boldsymbol{N}) \) has exactly \( \boldsymbol{6} \) distinct real roots then number of possible values of \( \boldsymbol{n} \) are :
- A. 15
- B. 16
- C. 17
- D. 14
Step-by-step solution
The function f(x)=x^3-12x. Consider t=|x|≥0, so equation becomes |f(t)|=n. For n∈N, analyze g(t)=|f(t)|. On [0,√12], g(t) increases from 0 to 16 at t=2 then decreases to 0. On [√12,∞), g(t) increases from 0. For 1≤n≤15, there are three t>0 solutions (two on (0,√12) and one on (√12,∞)), each giving two x=±t, total 6 distinct real roots. For n=16, only two t solutions giving 4 roots; for n>16, one t giving 2 roots. Hence possible n values are 1 to 15, total 15.
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