Exercise · ZW-MEAN-VALUE-THEOREM-008 · ★★★ Advanced
Find the value c guaranteed by the mean value theorem for \(f(x)=x^{3}-3x\) on \([0,2]\).
Choose one option.
Show answer
- A. \(\frac{2\sqrt{3}}{3}\)
Explanation
The secant has slope \(\frac{f(2)-f(0)}{2}=1\). From \(f'(c)=3c^{2}-3=1\) we get \(c^{2}=\frac{4}{3}\), and only \(c=\frac{2\sqrt{3}}{3}\) lies in \((0,2)\).
Review the concept: Mean value theorem
If \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\), there is a \(c\in(a,b)\) such that the slope of the tangent at \(c\) equals the slope of the secant line joining \((a,f(a))\) and \((b,f(b))\).
\[f'(c)=\frac{f(b)-f(a)}{b-a}\]
More practice problems: mean value theorem
- ZW-MEAN-VALUE-THEOREM-001 Find the value c guaranteed by the mean value theorem for f(x)=x^2 on [1,3].
- ZW-MEAN-VALUE-THEOREM-002 Find the value c guaranteed by the mean value theorem for f(x)=x^3 on [0,2].
- ZW-MEAN-VALUE-THEOREM-003 Find the value c guaranteed by the mean value theorem for f(x)=√x on [1,4].
- ZW-MEAN-VALUE-THEOREM-004 Which hypothesis does the mean value theorem require for f on [a,b]?
- ZW-MEAN-VALUE-THEOREM-005 Compute the slope of the secant line of f(x)=x^3 between x=1 and x=2.
- ZW-MEAN-VALUE-THEOREM-006 According to the mean value theorem, what slope must the tangent at the point c have?
- ZW-MEAN-VALUE-THEOREM-007 Find the value c guaranteed by the mean value theorem for f(x)=x^2-4x on [0,4].
- ZW-MEAN-VALUE-THEOREM-009 If f'(x)=0 for all x in (a,b) and f is continuous on [a,b], what can be said about f?