Let f(x) = max [x2,2-x2}, -2≤x≤2, then the function f(x)a)has a local maximum at x = 0b)has a local minima at x = -2c)has a global maximum at x = 2d)has local as well as global minima at x =Correct answer is option 'A,C,D'. Can you explain this answer?

Question

Let f(x) = max [x2,2-x2}, -2&le;x&le;2, then the function f(x)a)has a local maximum at x = 0b)has a local minima at x = -2c)has a global maximum at x = 2d)has local as well as global minima at x =Correct answer is option 'A,C,D'. Can you explain this answer?

Answer

So here x =0 be the critical point is [-2,2]
Clearly f'(0) = -2< 0
⇒ f has local maxima at x = 0
Now the sign of f'(x) change from negative to positive at x = -1 also at x = 1.
⇒ f has local minima’s at x = ±1.
Clearly x = ±1 are the global minimas also for the function.

⇒f is decreasing in this interval
Similarly f¹(x)> 0 in [1,2]
⇒ f is increasing in [1,2]
⇒ f has local maxima at x = 2.
⇒ global maxima = max { f ( 2 ) , f(-2) ,f(0)}
= max{4,4,2}
= 4
⇒ f has global maxima at x = ±2

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