1. **State the problem:** Show that the equation $$x^3 + 7x - 5 = 0$$ has a solution between $$x=0$$ and $$x=1$$. 2. **Evaluate the function at $$x=0$$:** $$f(0) = 0^3 + 7\times0 - 5 = -5$$ 3. **Evaluate the function at $$x=1$$:** $$f(1) = 1^3 + 7\times1 - 5 = 1 + 7 - 5 = 3$$ 4. **Apply the Intermediate Value Theorem:** Since $$f(0) = -5 < 0$$ and $$f(1) = 3 > 0$$, the function changes sign between $$x=0$$ and $$x=1$$. 5. **Conclusion:** By the Intermediate Value Theorem, there must be at least one solution to $$x^3 + 7x - 5 = 0$$ in the interval $$[0,1]$$. **Final answer:** There is a solution between $$x=0$$ and $$x=1$$.