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` .