Question

Given two curves: y=f(x) passing through the point (0,1) and g(x) = ∫ˣ₋[infinity]f(t) passing through the point (0, 1/n)The tangents drawn to both the curves at the points with equal abscissas intersect on the x-axis. Find the curve y=f(x)

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Answer 1

To find the curve y=f(x), we need to find the equations of the tangents to f(x) and g(x) at their respective points of tangency and solve for their intersection on the x-axis.

Step-by-step explanation:

To find the curve y=f(x), we need to use the given information about the tangents and intersections. Let's start by finding the derivative of g(x).

Since g(x) is an integral, we use the Fundamental Theorem of Calculus to differentiate it. So, g'(x) = f(x). Now, let's find the equation of the tangent to f(x) at the point (0,1) using the derivative. Since the slope of the tangent is f'(0), we have the equation of the tangent as y - 1 = f'(0)(x - 0).

Similarly, let's find the equation of the tangent to g(x) at the point (0, 1/n). Since the slope of the tangent is g'(0) = f(0), we have the equation of the tangent as y - 1/n = f(0)(x - 0). Now we need to find the intersection point of these two tangents. By setting the y-values equal and the x-values equal, we can solve for x.

Finally, substitute the x-value into the equation of the tangent to find the corresponding y-value. This will give us the curve y=f(x).