1. **State the problem:** We need to sketch the graph of the function $$f(x) = 3x^4 - 4x^3$$. 2. **Understand the function:** This is a polynomial function of degree 4. The general shape of a quartic function depends on the leading coefficient and the roots. 3. **Find the roots (x-intercepts):** Set $$f(x) = 0$$: $$3x^4 - 4x^3 = 0$$ Factor out $$x^3$$: $$x^3(3x - 4) = 0$$ So, $$x^3 = 0$$ or $$3x - 4 = 0$$ This gives roots: $$x = 0$$ (with multiplicity 3) and $$x = \frac{4}{3}$$. 4. **Find the y-intercept:** Evaluate $$f(0)$$: $$f(0) = 3(0)^4 - 4(0)^3 = 0$$ So the graph passes through the origin. 5. **Find critical points (extrema):** Compute the derivative: $$f'(x) = 12x^3 - 12x^2 = 12x^2(x - 1)$$ Set $$f'(x) = 0$$: $$12x^2(x - 1) = 0$$ So critical points at $$x = 0$$ and $$x = 1$$. 6. **Evaluate $$f(x)$$ at critical points:** - At $$x=0$$: $$f(0) = 0$$ - At $$x=1$$: $$f(1) = 3(1)^4 - 4(1)^3 = 3 - 4 = -1$$ 7. **Determine the nature of critical points:** Compute second derivative: $$f''(x) = 36x^2 - 24x = 12x(3x - 2)$$ - At $$x=0$$: $$f''(0) = 0$$ (inconclusive) - At $$x=1$$: $$f''(1) = 12(1)(3 - 2) = 12 > 0$$, so $$x=1$$ is a local minimum. 8. **Behavior at infinity:** Since the leading term is $$3x^4$$, as $$x \to \pm \infty$$, $$f(x) \to +\infty$$. 9. **Summary:** - Roots at $$x=0$$ (multiplicity 3) and $$x=\frac{4}{3}$$. - Local minimum at $$x=1$$ with $$f(1) = -1$$. - Passes through origin. - Ends go to positive infinity. This information helps sketch the graph accurately.