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In proving that $Vertcdot Vert$ defined on $C^{1}[a,b]$ by $Vert{f Vert}=maxlimits_{aleq tleq b}left|f(t)right|+maxlimits_{aleq tleq b}left|dfrac{d}{dt}f(t)right|$ is a norm, I encountered a problem. It's getting the equation to satisfy the first condition.

MY WORK

Let $fin C^{1}[a,b],$ then

begin{align} Vert{f Vert}=0 &leftrightarrow maxlimits_{aleq tleq b}left|f(t)right|+maxlimits_{aleq tleq b}left|dfrac{d}{dt}f(t)right|=0 \& leftrightarrow left|f(t)right|+left|dfrac{d}{dt}f(t)right|=0,;;tin [a,b] \& leftrightarrow f(t)+dfrac{d}{dt}f(t)=0,;;tin [a,b]\& leftrightarrow f(t)=e^{-t},;;tin [a,b]end{align}
I'm not getting $f(t)=0,;;forall,tin [a,b].$ Please, where did I get it wrong? Can someone help me? As to the other two conditions, I have no problems with them.

You dropped the absolute value bars between the second and third lines. If you have $|f(t)| + left| frac{d}{dt} f(t) right| = 0$ for all $t in [a,b]$, then it must be the case that both $f(t) = 0$ and $frac{d}{dt} f(t) = 0$. Really, the first suffices, as this gives $f(t) = 0$ for all $t in [a,b]$.