Final answer:
To find G'(a), use the power rule of differentiation and substitute a for x in the derivative. The equation of the tangent lines to the curve y = 5x^2 - x^3 at the points (3, 18) and (4, 16) can be found using the point-slope form with the slope given by G'(a).
Step-by-step explanation:
To find the derivative G'(a) of the function G(X) = 5x^2 - x^3, we need to use the power rule of differentiation. The power rule states that the derivative of x^n, where n is a constant, is equal to n*x^(n-1). Applying this rule to each term of the function, we get G'(X) = 10x - 3x^2. To find G'(a), we substitute a for x in the derivative, so G'(a) = 10a - 3a^2.
To find the equation of the tangent line to the curve y = 5x^2 - x^3 at the point (3, 18), we can use the slope-intercept form of a linear equation, y = mx + b. The slope of the tangent line is equal to G'(a), which is 10a - 3a^2. Using the point-slope form, we have y - y1 = m(x - x1), where (x1, y1) is the given point. Substituting the values (3, 18), we get y - 18 = (10a - 3a^2)(x - 3). This is the equation of the tangent line passing through (3, 18). Similarly, we can find the equation of the tangent line passing through (4, 16) by substituting the values (4, 16) into the point-slope form equation.