The equation of the tangent at (2,3) on the curve `y^2=a x^3+b` is `y=4x-5.` Find the values of a and b

Explanation

To solve the problem, we need to find the values of \( a \) and \( b \) for the curve given by the equation \( y^2 = ax^3 + b \), knowing that the tangent at the point \( (2, 3) \) is given by the equation \( y = 4x - 5 \).

### Step 1: Identify the slope of the tangent
The slope of the tangent line \( y = 4x - 5 \) is the coefficient of \( x \), which is \( 4 \).

### Step 2: Differentiate the curve equation
To find the slope of the tangent to the curve \( y^2 = ax^3 + b \), we differentiate both sides with respect to \( x \):
\[
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