Question 76
What is the derivative of $f(x) = x^76$ at x = 1?
Detailed Solution
f'(x) = 76 $x^{75}$. At x = 1, f'(1) = 76.
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Question 77
Consider the function f(x) = (x - $77)^2$. Which of the following statements are TRUE?
Detailed Solution
f(77) = 0, f'(77) = 0, f''(77) = 2 > 0 (so x = 77 is a local MINIMUM, not maximum).
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Question 78
Find the value of $\lim_{x \to 78} \frac{x^2 - 78^2}{x - 78}$.
Detailed Solution
Using L'Hopital's rule or factoring: $(x^2 - 78^2)/(x - 78)$ = x + 78. At x = 78, limit = 156.
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Question 79
What is the derivative of $f(x) = x^79$ at x = 1?
Detailed Solution
f'(x) = 79 $x^{78}$. At x = 1, f'(1) = 79.
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Question 80
Consider the function f(x) = (x - $80)^2$. Which of the following statements are TRUE?
Detailed Solution
f(80) = 0, f'(80) = 0, f''(80) = 2 > 0 (so x = 80 is a local MINIMUM, not maximum).
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