1. Problem 25: Show that the equation $x^3 - 15x + c = 0$ has at most one solution in the interval $[-2,2]$. 2. We use the fact that if a function is strictly monotonic (always increasing or always decreasing) on an interval, it can have at most one root there. 3. Consider the function $f(x) = x^3 - 15x + c$. Its derivative is $f'(x) = 3x^2 - 15$. 4. On the interval $[-2,2]$, evaluate $f'(x)$: $$f'(x) = 3x^2 - 15 \\ \text{Since } x^2 \leq 4, \quad 3x^2 \leq 12 \\ \Rightarrow f'(x) \leq 12 - 15 = -3 < 0$$ 5. Because $f'(x) < 0$ for all $x$ in $[-2,2]$, $f$ is strictly decreasing on this interval. 6. A strictly decreasing continuous function can cross zero at most once, so $f(x) = 0$ has at most one solution in $[-2,2]$. --- 7. Problem 26: Show that the equation $x^4 + 4x + c = 0$ has at most two real solutions. 8. Let $g(x) = x^4 + 4x + c$. Its derivative is $g'(x) = 4x^3 + 4$. 9. To find the number of real roots, analyze $g'(x) = 0$: $$4x^3 + 4 = 0 \\ x^3 = -1 \\ x = -1$$ 10. $g'(x)$ changes sign only once at $x = -1$, so $g(x)$ has exactly one critical point. 11. A polynomial with one critical point can have at most two real roots because the graph can cross the x-axis at most twice. --- 12. Problem 27(a): Show that a polynomial of degree 3 has at most three real zeros. 13. By the Fundamental Theorem of Algebra, a degree 3 polynomial has exactly 3 roots (counting multiplicities and complex roots). 14. Since complex roots come in conjugate pairs, the number of real roots is at most 3. --- 15. Problem 27(b): Show that a polynomial of degree $n$ has at most $n$ real zeros. 16. By the Fundamental Theorem of Algebra, a degree $n$ polynomial has exactly $n$ roots (counting multiplicities and complex roots). 17. Therefore, the number of real zeros cannot exceed $n$. --- 18. Problem 28(a): Suppose $f$ is differentiable on $\mathbb{R}$ and has two zeros. Show that $f'$ has at least one zero. 19. By Rolle's Theorem, if $f(a) = f(b) = 0$ for $a < b$, then there exists $c \in (a,b)$ such that $f'(c) = 0$. 20. Since $f$ has two zeros, $f'$ must have at least one zero between them. --- 21. Problem 28(b): Suppose $f$ is twice differentiable on $\mathbb{R}$ and has three zeros. Show that $f''$ has at least one real zero. 22. By applying Rolle's Theorem twice: - Since $f$ has three zeros, $f'$ has at least two zeros. - Applying Rolle's Theorem to $f'$, $f''$ has at least one zero. --- 23. Problem 28(c): Generalization. 24. If $f$ is $k$ times differentiable and has $k+1$ zeros, then $f^{(k)}$ has at least one zero. 25. This follows by applying Rolle's Theorem repeatedly $k$ times. --- 26. Problem 29: If $f(1) = 10$ and $f'(x) \geq 2$ for $1 \leq x \leq 4$, how small can $f(4)$ possibly be? 27. Since $f'(x) \geq 2$, the function is increasing at a rate of at least 2 on $[1,4]$. 28. By the Mean Value Theorem: $$f(4) - f(1) = f'(c)(4 - 1) \geq 2 \times 3 = 6$$ 29. Therefore: $$f(4) \geq f(1) + 6 = 10 + 6 = 16$$ 30. The smallest possible value of $f(4)$ is 16. 31. We suppose $f'(x) \geq 2$ to ensure the function grows at least linearly with slope 2, so the increase from $x=1$ to $x=4$ is at least 6.