Exercise · ZW-MEAN-VALUE-THEOREM-003 · ★★☆ Intermediate
Find the value c guaranteed by the mean value theorem for \(f(x)=\sqrt{x}\) on \([1,4]\).
Choose one option.
Show answer
- B. \(\frac{9}{4}\)
Explanation
The slope of the secant is \(\dfrac{f(4)-f(1)}{4-1}=\frac{1}{3}\). Solve \(f'(c)=\frac{1}{3}\) with \(f'(x)=\frac{1}{2 \sqrt{x}}\) and the solution inside \((1,4)\) is c = \frac{9}{4}.
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-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-008 Find the value c guaranteed by the mean value theorem for f(x)=x^3-3x on [0,2].
- 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?