`f(x) = cos(x) + tan(x), [0,pi]` Determine whether the Mean Value Theorem can be applied to `f` on the closed interval `[a,b]`. If the Mean Value Theorem can be applied, find all values of `c` in the open interval `(a,b)` such that `f'(c) = (f(b) - f(a))/(b - a)`. If the Mean Value Theorem cannot be applied, explain why not.
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Mean value theorem can be applied,
1. if f is continuous on the closed interval `[a,b]` ,
2. if f is differentiable on the open interval (a,b)
3. there is a number c in (a,b) such that
Let us check the continuity of the function on the closed interval `[0,pi]` by plotting the graph of the function.
So the function is not continuous at x=`pi` /2
f(pi/2) is undefined
So the function is differentiable.
Now there is no value of c in the closed interval `[0,pi]` for which f'(c) is negative.
So the function does not satisfies the condition number 1 and 3. Hence Mean value theorem can not be applied.
Graph is attached Blue colour is for function and red for f'(x)
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