Let f(x) and g(x) be functions with domain \((0,\infty) \) . Suppose \(f(x)=x^2 \) and the tangent line to f(x) at x=a is perpendicular to the tangent line to g(x) at x=a for all positive real numbers a. Find all possible functions g(x).
Sei f (x) und g (x) Funktionen mit Domäne \((0,\infty) \). \(f(x)= x^2 \) und die Tangentenlinie zu f (x) sei bei x = a für alle positiven reellen Zahlen a senkrecht zur Tangentenlinie zu g (x). Finde alle möglichen Funktionen g (x).
Hello yeliah!
\(f(x)=x^2\\ f'(x)=2x\)
\(g'(x)=-\frac{1}{f'(x)}\\ g'(x)=-\frac{1}{2x}\)
\(g(x)=-\frac{1}{2}\cdot \int x^{-1}\cdot dx\)
\(g(x)=-\frac{1}{2}\cdot ln |x|+C\)
\(\{possible\ g(x)\}\in\{(-\frac{1}{2}\cdot ln |\mathbb Q|+\mathbb Q)\}\) Is this the correct answer?
[Added by Melody: No, this isn't right. Look at my next post for other points]
The tangent line to f(x) at x=a \(\large ?\) is perpendicular to the tangent line to g(x) at x=a \(\large ?\) for all positive real numbers a. \(\large ?\) Find all possible functions g(x).
I do not understand this text. Is a the same as | x |? Why?
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+118667 
