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Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
a- if f is a function of x and y and a is a real number, then f(ax, ay)= af(x,y).
b- if fx(a,b) < 0, then f is decreasing with respect to x near (a,b).

  • calculus -

    (a) is certainly not true in general. While it is true for linear functions, it is otherwise not true.

    f(x,y) = x^2/(y^2+1)
    f(ax,ay) = (a^2x^2)/(a^y^2+1)


    f(x,y) = sin(x+y)
    f(ax,ay) = sin a(x+y)


    f(x,y) = e^x * ln(y)
    f(ax,ay) = a^ax * ln(ay) = e^a * e^x * (lna + lny)

    (b) I assume fx means the partial with respects to x. That's true. The derivative is the slope of the curve in the intersection of f(x,y) and the plane y=b.

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