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).
(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)
or
f(x,y) = sin(x+y)
f(ax,ay) = sin a(x+y)
or
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.
a- To determine if the statement is true or false, we can apply the property of linearity of functions. If the function f is linear, then the statement is true. However, if f is not a linear function, then the statement is false.
Let's consider a counterexample to show that the statement is false. Suppose we have a function f(x, y) = x^2 + y^2 and let a = 2. If we substitute ax and ay into the function, we get f(2x, 2y) = (2x)^2 + (2y)^2 = 4x^2 + 4y^2. However, af(x, y) = 2(x^2 + y^2) = 2x^2 + 2y^2. As we can see, f(2x, 2y) is not equal to af(x, y), which means the statement is false.
b- To determine if the statement is true or false, we need to understand what it means for a function to be decreasing with respect to x near a certain point (a,b). If the function fx(a,b) < 0, it means that the derivative of the function with respect to x is negative at the point (a,b). Therefore, if the derivative of the function remains negative near the point (a,b), then the statement is true.
To provide an explanation of why it is true, we can use the concept of derivative. If the derivative of a function with respect to x is negative on an interval around the point (a,b), it indicates that the function is decreasing on that interval. Therefore, if fx(a,b) < 0, this suggests that the function is decreasing with respect to x near the point (a,b). This is why the statement is true.
It's important to note that these explanations rely on mathematical concepts and properties, so a thorough understanding of calculus and function properties is required to arrive at these conclusions.