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Let A be the area of the triangle formed by x-axis, y-axis and the tangent line of $y=x^p$ ($p<0$) at $x=c$. Find $p$ such that $A$ is independent of $c$.

P.S What does "$A$ is independent of $c$" mean?

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You should get $p=-1$. – Mhenni Benghorbal Oct 21 '12 at 9:10
up vote 2 down vote accepted

Hint: Find the first derivative $f'$ of $f\colon x\mapsto x^p$ and using it the exact tangent line equation $y=mx+b$ through the point $(c,f(c))$. Find the intercepts $(x_0,0)$ and $(0,y_0)$ of that line. Then the area is $A=\frac 12 x_0 y_0$. This will be an expression that depends on the givens $c$ and $p$. For some $p$, the expression can be simplified in such a manner that it does not involve $c$ at all, i.e. for that $p$, $A$ is independent of $c$.

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