Try to realize that \( f(x) = x^2 - 2|x| \). Try using the graph. If you still don't get, reply I'll add a detailed solution. Just to check are the answers C, D , D ?
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Sudeep Salgia
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2 years, 1 month ago

Here is the image for the graphs in the question. In the first part of \(g(x) \), the value of \(g(x)\) is given by the minimum value of \( f \) in the range \( (-3 ,x) \). Since the function is decreasing till \( -1 \), \(g(x) \) matches with \( f(x) \). After that the value of \( g(x) \) becomes a constant equal to \( -1 \) since that is the local minima. Similarly, we can plot it for the positive part. If it still sounds unclear, let me know I'll elaborate further. The answer to the third question should be C.
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Sudeep Salgia
·
2 years, 1 month ago

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@Sudeep Salgia
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Why did you graphed a straight line in \((1,2)\) , I was thinking of it to be \(f(x)\).
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Kushal Patankar
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2 years, 1 month ago

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@Kushal Patankar
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It is maximum value of \(f(t) \) where \( 0 \leq t \leq x\). Since the function is decreasing, the value would be \(f(0) \) which is \(0\).
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Sudeep Salgia
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2 years, 1 month ago

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@Sudeep Salgia
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But in \((1,2)\) it is increasing. So all \(f(x)\) will be greater than \(f(t)\).
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Kushal Patankar
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2 years, 1 month ago

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@Kushal Patankar
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Sorry for my misleading statement. It is true that the function is increasing in the interval \((1,2)\) but notice that the value is still negative, that is, \( f(r) < f(0) = 0 \ \ \forall \ r \ \ \in (1,2) \). Since \( t \ \in [0, x] \), therefore \( \forall \ x < 2 , f(0) > f(x) \) .
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Sudeep Salgia
·
2 years, 1 month ago

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TopNewestTry to realize that \( f(x) = x^2 - 2|x| \). Try using the graph. If you still don't get, reply I'll add a detailed solution. Just to check are the answers C, D , D ? – Sudeep Salgia · 2 years, 1 month ago

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– Kushal Patankar · 2 years, 1 month ago

I am unable to graph \(g(x)\).Log in to reply

img

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– Kushal Patankar · 2 years, 1 month ago

Why did you graphed a straight line in \((1,2)\) , I was thinking of it to be \(f(x)\).Log in to reply

– Sudeep Salgia · 2 years, 1 month ago

It is maximum value of \(f(t) \) where \( 0 \leq t \leq x\). Since the function is decreasing, the value would be \(f(0) \) which is \(0\).Log in to reply

– Kushal Patankar · 2 years, 1 month ago

But in \((1,2)\) it is increasing. So all \(f(x)\) will be greater than \(f(t)\).Log in to reply

– Sudeep Salgia · 2 years, 1 month ago

Sorry for my misleading statement. It is true that the function is increasing in the interval \((1,2)\) but notice that the value is still negative, that is, \( f(r) < f(0) = 0 \ \ \forall \ r \ \ \in (1,2) \). Since \( t \ \in [0, x] \), therefore \( \forall \ x < 2 , f(0) > f(x) \) .Log in to reply

– Kushal Patankar · 2 years, 1 month ago

OK, got you, thanks ☺☺☺Log in to reply

– Sudeep Salgia · 2 years, 1 month ago

Happy to help. :)Log in to reply