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Olinguito
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If the line $x=k$ is tangent to the curve
$$\large y\:=\:x+\sqrt2\,e^{\frac{x+y}{\sqrt2}}$$
what is the value of $k$?
$$\large y\:=\:x+\sqrt2\,e^{\frac{x+y}{\sqrt2}}$$
what is the value of $k$?
Olinguito said:If the line $x=k$ is tangent to the curve
$$\large y\:=\:x+\sqrt2\,e^{\frac{x+y}{\sqrt2}}$$
what is the value of $k$?
Olinguito said:If the line $x=k$ is tangent to the curve
$$\large y\:=\:x+\sqrt2\,e^{\frac{x+y}{\sqrt2}}$$
what is the value of $k$?
The curve represented by this equation is an exponential curve with a base of e raised to the power of x+y divided by the square root of 2, added to the value of x plus the square root of 2.
When x=k is tangent to a curve, it means that the line passing through the point (k, f(k)) is perpendicular to the tangent line of the curve at that point. In other words, the slope of the line passing through (k, f(k)) is equal to the slope of the tangent line at that point.
To find the value of k, we can use the derivative of the curve and set it equal to the slope of the line passing through (k, f(k)). This will give us a system of equations which we can solve to find the value of k.
Finding the value of k allows us to determine the point at which the curve has a unique tangent line. This can be useful in solving optimization problems or finding the minimum or maximum values of a function.
Yes, there can be more than one value of k for which x=k is tangent to the curve. This occurs when the curve has a horizontal tangent line at multiple points, meaning the slope of the curve is equal to 0 at those points.