Mean Value/Rolle's Theorem

• April 28th 2009, 05:05 PM
Zocken
Mean Value/Rolle's Theorem
Consider the function $f(x)=6x^4-7x+1$ on the interval $[1/2,2].$ $f$ has at least one zero in $[1/2,2]$ (actually f(1)=0) so show that $f$ has *exactly* on zero in the indicated interval. Hint:Use the Mean Value Theorem/Rolle's Theorem to prove the result.
• April 28th 2009, 05:22 PM
NonCommAlg
Quote:

Originally Posted by Zocken
Consider the function $f(x)=6x^4-7x+1$ on the interval $[1/2,2].$ $f$ has at least one zero in $[1/2,2]$ (actually f(1)=0) so show that $f$ has *exactly* on zero in the indicated interval. Hint:Use the Mean Value Theorem/Rolle's Theorem to prove the result.

if $f$ had more than one zero in [1/2, 2], then by Rolle's theorem $f'(x)=0$ would have at least one root in [1/2, 2]. but the (real) root of $f'(x)=24x^3-7=0$ is $x=\sqrt[3]{7/24} \notin [1/2 , 2].$
• April 28th 2009, 05:24 PM
Zocken
wow makes it so simple. thanks a bunch!