Let f(x)= begincases x3-x2+10 x-7, x ≤ 1 -2 x+ log 2(b2-4), x1 endcases Then the set of all values of b, for which f(x) has maximum value at x=1, is :

Question

Let f(x)= begincases x3-x2+10 x-7, x ≤ 1 -2 x+ log 2(b2-4), x>1 endcases Then the set of all values of b, for which f(x) has maximum value at x=1, is : (A) (-6,-2) (B) (2,6) (C) [-6,-2) ∪(2,6] (D) [-√6,-2) ∪(2, √6]

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f (1)=3 For x < 1, f prime( x )=3 x 2-2 x +10>0 ⇒ f ( x ) is increasing For x >1, f prime( x ) < 0 ⇒ function is decreasing. displaystyle lim x arrow 1+ f ( x )=-2+ log 2( b 2-4) For maximum value at x =1 3 ≥-2+ log 2( b 2-4) 32 ≥ b 2-4>0 b ∈[-6,-2) ∪(2,6]

Q. Let $f (x) = {x^{3} - x^{2} + 10 x - 7, - 2 x + lo g_{2} (b^{2} - 4), x \leq 1 x > 1$
Then the set of all values of $b$ , for which $f (x)$ has maximum value at $x = 1$ , is :

$(- 6, - 2)$

$(2, 6)$

$[- 6, - 2) \cup (2, 6]$

$[- 6, - 2) \cup (2, 6]$

Q. Let f(x)={x3−x2+10x−7,−2x+log2​(b2−4),​x≤1x>1​ Then the set of all values of b, for which f(x) has maximum value at x=1, is :

(−6,−2)

(2,6)

[−6,−2)∪(2,6]

[−6​,−2)∪(2,6​]

f(1)=3 For x<1,f′(x)=3x2−2x+10>0 ⇒f(x) is increasing For x>1,f′(x)<0 ⇒ function is decreasing. x→1+lim​f(x)=−2+log2​(b2−4) For maximum value at x=1 3≥−2+log2​(b2−4) 32≥b2−4>0 b∈[−6,−2)∪(2,6]

Q. Let $f (x) = {x^{3} - x^{2} + 10 x - 7, - 2 x + lo g_{2} (b^{2} - 4), x \leq 1 x > 1$
Then the set of all values of $b$ , for which $f (x)$ has maximum value at $x = 1$ , is :

$(- 6, - 2)$

$(2, 6)$

$[- 6, - 2) \cup (2, 6]$

$[- 6, - 2) \cup (2, 6]$