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  • if f(x) is a double differentiable function such that |f''(x)|

if f(x) is a double differentiable function such that |f''(x)| ? 5 for each x in the interval [0,4] and f takes it largest value at an interior point of this interval,then the maximum possible value of |f'(0)|+|f"(4)| is explain briefly

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@raj ( 7016747672 )

if f is taking its maximum value in [0,4] implies f'(x)=0 at that point, lets say at point x = c.

now using rolles theorum, we can say

 

[f'(c)-f'(0)]/\ (c-0) = f''(m) \ for \ some \ m \\\\where\\ m \varepsilon [0,4]

now, f'(c) = 0 implies \left|f'\left(0\right)\right|\:=\:c\left|f''\left(m\right)\right|

so to maximise |f'(0)|+|f"(4)| = c|f''(m)| + |f''(4)| which is less than or equal to c*5 + 5 if we put c=4 which is a possible value of c which means

the answer is  25.

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Kshitij

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