1. Problem: Find the value of $f'(3)$ for $f(x)=x^5+6x+4$. 2. Differentiate $f(x)$: $$f'(x)=5x^4+6$$ 3. Evaluate at $x=3$: $$f'(3)=5(3)^4+6=5(81)+6=405+6=411$$ Answer: 411 (option d). 4. Problem: Is the statement "A function is increasing on an interval if its derivative is negative on that interval" true? 5. Explanation: A function is increasing if its derivative is positive, not negative. Answer: False (option a). 6. Problem: If $f'(x) > 0$ at a critical point, does the function have a relative maximum there? 7. Explanation: At a critical point, if $f'(x) > 0$, the function is increasing, so it cannot be a relative maximum. Answer: False (option b). 8. Problem: Which statement is true if $f'(x) > 0$ for all $x$ in interval $I$? 9. Explanation: Positive derivative means function is increasing. Answer: $f(x)$ is increasing on $I$ (option d). 10. Problem: Find the equation of the tangent line to $f(x)=x^3-2x$ at $(1,-1)$. 11. Differentiate: $$f'(x)=3x^2-2$$ 12. Evaluate slope at $x=1$: $$f'(1)=3(1)^2-2=3-2=1$$ 13. Use point-slope form: $$y - (-1) = 1(x - 1) \\ y + 1 = x - 1 \\ y = x - 2$$ Answer: $y=x-2$ (option b). 14. Problem: Is a monotonic function both increasing and decreasing? 15. Explanation: Monotonic means either entirely non-increasing or non-decreasing, not both. Answer: False (option b). 16. Problem: Is a point where $f'(x)=0$ and changes sign called a relative extremum? 17. Explanation: Yes, such points are relative maxima or minima. Answer: True (option a). 18. Problem: Find the point of tangency where slope of tangent to $y=x^2-4x+5$ at $x=2$ is zero. 19. Differentiate: $$y' = 2x - 4$$ 20. Evaluate slope at $x=2$: $$y'(2) = 2(2) - 4 = 4 - 4 = 0$$ 21. Find $y$ at $x=2$: $$y(2) = 2^2 - 4(2) + 5 = 4 - 8 + 5 = 1$$ Answer: $(2,1)$ (option d). 22. Problem: Find $f'(x)$ for $f(x)=x^6 - 2x^5 + 3x^2 - 23$. 23. Differentiate term-wise: $$f'(x) = 6x^5 - 10x^4 + 6x$$ Answer: $6x^5 - 10x^4 + 6x$ (option d). 24. Problem: Find the derivative of $f(x) = x^5$. 25. Derivative: $$f'(x) = 5x^4$$ Answer: $5x^4$ (option a).