Idk how to do this question please help

Question

Please help solve this question:

Find the maximum area of a triangle formed in the first quadrant by the x-axis, y-axis and a tangent line to the graph of f=(x+7)^-2

Answer 1

A wonderful question.

So, first things first, let's say our tangent line intersects the graph (x+7)^-2 at point x=t.

f(t) = (t+7)^-2, f'(t) = -2(t+7)^-3

So our tangent line would be -2(x-t)/(t+7)³ + (t+7)^-2

Now, the area of the triangle formed by this line, along with the x and y axis.

y intercept: x=0, y=2t/(t+7)³ + (t+7)^-2 = (3t+7)/(t+7)³

x intercept: y=0

x-t = - ((t+7)^-2) / (-2/(t+7)³) = (t+7)/2

x = (3t+7)/2

The area is therefore 1/2 * (3t+7)/(t+7)³ * (3t+7)/2

The area is (3t+7)²/(4(t+7)³)

This is minimized either on the boundaries (t=0 or t=∞) or at a point where the derivative is 0.

t=0: 7²/(4*7³) = 1/28

t=∞: Numerator is degree 2, denominator is degree 3, area is 0.

d/dt (3t+7)²/(4(t+7)³) = (2*3*(3t+7)*(t+7)³ - (3t+7)²*3(t+7)²) / (4(t+7)⁶) = 3(3t+7)(t+7)²(2(t+7)-(3t+7))/(4(t+7)⁶) = 3(3t+7)(-t+7)/(4(t+7)⁴)

t = 7 or t = -7/3

t cannot be -7/3 since that's out of bounds

Area = (28)²/(4*14³) = 2²/(4*14) = 1/14

1/14 is bigger than 1/28

So the maximum area of this triangle is 1/14.

And it occurs when the tangent line intersects the point (7, 1/196)

Answer 2

desmos.com/calculator/pxfupqf1jf

Visit the above graph and use slider on x1 to see different triangles along with respective area.