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Derivatives – test 1: The theorems of Fermat, Rolle, Lagrange and Cauchy

10 multiple-choice and short-answer problems on derivatives (Mathematics I). The solutions are shown after checking.

For multiple-choice questions choose one answer. For the others type the answer: a number, a decimal or a fraction such as 5/12. When you finish, click Check answers.

1. Find \(\xi\) from Rolle’s theorem for \(f(x)=x^2-6x+5\) on \([1,5]\).

2. Which \(\xi\) does Rolle’s theorem give for \(f(x)=x^3-4x\) on \([0,2]\)?

3. Which function satisfies the conditions of Rolle’s theorem on \([-1,1]\)?

4. Find \(\xi\) from Lagrange’s theorem for \(f(x)=x^3\) on \([0,3]\).

5. Which \(\xi\) does Lagrange’s theorem give for \(f(x)=\ln x\) on \([1,e]\)?

6. Find \(\xi\) from Lagrange’s theorem for \(f(x)=\sqrt x\) on \([1,4]\).

7. Find \(\xi\) from Cauchy’s theorem for \(f(x)=x^2\), \(g(x)=x^3\) on \([1,2]\).

8. Why does Rolle’s theorem not apply to \(f(x)=|x-2|\) on \([1,3]\)?

9. Lagrange’s theorem proves \(|\sin x-\sin y|\le k|x-y|\) for all \(x,y\). What is the smallest such \(k\)?

10. The equation \(3^x+4^x=5^x\) has exactly one solution. Which?