Explain how show that given function f with domain [0,1] and`f(x)=6x^3+4x-9`  have a value a in (0,1) so f(a)=0 and no solve equations?

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Since the problem does not request the solutions to the given equation `6x^3+4x-9 = 0`  but it requests to check if there exists a solution in the given interval `[0,1], ` you need to evaluate the values of the function at the limits of interval, `x = 0`  and `x...

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Since the problem does not request the solutions to the given equation `6x^3+4x-9 = 0`  but it requests to check if there exists a solution in the given interval `[0,1], ` you need to evaluate the values of the function at the limits of interval, `x = 0`  and `x = 1`  such that:

`f(0) = -9`

`f(1) = 6 + 4 - 9 = 1`

Notice that`f(0) = -9 < 0`  and `f(1) = 1 > 0` , hence, since the polynomial function is continuous over `[0,1]`  then, for a value `a in [0,1], ` the graph of function intersects x axis such that `f(a) = 0` .

Hence, evaluating the values of the continuous function at the limits of interval yields that there exists a value `a in [0,1]`  such that `f(a) = 0` . 

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