If f(x) is differentiable function wherever it is continuous and f'(c1)=f'(c2)=0,f''(c1).f''(c2) 0, then minimum number of roots of f'(x)=0 in [c1−1,c2+1] is

f | |(c1)f | |(c2) and f |(c1)=f |(c2)=0 ⇒f | |(c1)−f | |(c2)>0 ⇒f | |(c1)>0 and ⇒f | |(c2)<0 ⇒f |(x)=0 atleast four times in [c1−1,c2+1]

If f(x) is differentiable function wherever it is continuous and f'(c1)=f'(c2)=0,f''(c1).f''(c2) 0, then minimum number of roots of f'(x)=0 in [c1−1,c2+1] is

f | |(c1)f | |(c2) and f |(c1)=f |(c2)=0 ⇒f | |(c1)−f | |(c2)>0 ⇒f | |(c1)>0 and ⇒f | |(c2)<0 ⇒f |(x)=0 atleast four times in [c1−1,c2+1]

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If $f (x)$ is differentiable function wherever it is continuous and $f' (c_{1}) = f' (c_{2}) = 0, f'' (c_{1}) . f'' (c_{2}) < 0$ , $f (c_{1}) = 5, f (c_{2}) = 0$ and $(c_{1} < c_{2})$ .
If $f (x)$ is continuous in $[c_{1}, c_{2}]$ and $f'' (c_{1}) - f'' (c_{2}) > 0$ , then minimum number of roots of $f' (x) = 0$ in $[c_{1} - 1, c_{2} + 1]$ is

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Detailed Solution

$f^{| |} (c_{1}) f^{| |} (c_{2}) a n d f^{|} (c_{1}) = f^{|} (c_{2}) = 0 \Rightarrow f^{| |} (c_{1}) - f^{| |} (c_{2}) > 0 \Rightarrow f^{| |} (c_{1}) > 0 a n d \Rightarrow f^{| |} (c_{2}) < 0 \Rightarrow f^{|} (x) = 0 a t l e a s t f o u r t i m e s i n [c_{1} - 1, c_{2} + 1]$