1. The problem is to analyze the function $y=3x^4-\cos x$. 2. This function combines a polynomial term $3x^4$ and a trigonometric term $-\cos x$. 3. To understand its behavior, we can find its derivative: $$y' = \frac{d}{dx}(3x^4) - \frac{d}{dx}(\cos x) = 12x^3 + \sin x$$ 4. The critical points occur where $y' = 0$, i.e., where: $$12x^3 + \sin x = 0$$ 5. This equation is transcendental and may require numerical methods to solve exactly. 6. The function's intercepts: - At $x=0$, $y=3(0)^4 - \cos 0 = 0 - 1 = -1$, so the y-intercept is $(0,-1)$. - For x-intercepts, solve $3x^4 - \cos x = 0$; this also requires numerical methods. 7. The function grows rapidly for large $|x|$ due to the $3x^4$ term. Final answer: The function is $y=3x^4 - \cos x$ with derivative $y' = 12x^3 + \sin x$, y-intercept at $(0,-1)$, and critical points satisfying $12x^3 + \sin x = 0$.