greatest slope of tangent line

Question

Where on the curve y=(484+x^2)^-1  does the tangent line have the greatest slope?

Use the notation (a,b) to give the point

Answer 1

y' = -1(484 + x²)^-2·2x = -2x/(484 + x²)². We have to find maximum of this function.

y'' = -2[(484 + x²)² - x·2(484 + x²)·2x]/(484 + x²)⁴ = -2(484 + x² - 4x²)/(484 + x²)³ = -2(484 - 3x²)/(484 + x²)³

At maximum value of y' y'' = 0. So, 484 - 3x² = 0, x = ±22/√ 3.

So y = 3/1936. y' has maximum at x = - 22/√3 and y = 3/1936

(-22/√3, 3/1936) at this point we have maximum slope.