    Question

# Consider f(x)=x21+x2 and g(x)=px where p∈R. Then which of the following statements is (are) CORRECT?

A
If f(x)=g(|x|) has exactly three distinct solutions, then number of all possible real values of p is 1.
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B
If f(x)=g(|x|) has exactly five distinct solutions, then range of all possible real values of p is (0,12).
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C
Number of distinct tangents that can be drawn to the curve y=f(x) from the origin is 3.
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D
Number of distinct tangents that can be drawn to the curve y=f(x) from the origin is 1.
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Solution

## The correct options are A If f(x)=g(|x|) has exactly three distinct solutions, then number of all possible real values of p is 1. B If f(x)=g(|x|) has exactly five distinct solutions, then range of all possible real values of p is (0,12). C Number of distinct tangents that can be drawn to the curve y=f(x) from the origin is 3.f(x)≥0 limx→∞f(x)=1 So, range of f is [0,1) Clearly, f(x)=g(|x|) has x=0 as one solution. f(x)=g(|x|) has three solutions iff each branch of g(|x|)=p|x|, p>0 is tangent to y=f(x) ⇒f′(x)=p ⇒2x(1+x2)2=p Intersection point of f(x)=g(|x|): x21+x2=2x2(1+x2)2 ⇒1+x2=2 ⇒x=±1 ⇒p=12 as p>0 If p>12, y=f(x) will not intersect y=g(|x|) and hence, there will be no solution. For five distinct solutions, p∈(0,12)  Suggest Corrections  0      Similar questions  Explore more