Final answer:
To find the slope of the tangent line at x=4 for the function f(x) = -2x² - 4√(x), we find the derivative of f(x) and evaluate it at x=4. The slope of the tangent line at x=4 is -17. The equation of the tangent line at x=4 is y = -17x + 96.
Step-by-step explanation:
To find the slope of the tangent line at x=4 for the function f(x) = -2x² - 4√(x), we need to find the derivative of f(x) and evaluate it at x=4.
The derivative of f(x) can be found using the power rule and chain rule. The derivative is f'(x) = -4x - 2/√(x).
Evaluating f'(x) at x=4, we get f'(4) = -4(4) - 2/√(4) = -16 - 2/2 = -16 - 1 = -17.
Therefore, the slope of the tangent line at x=4 is -17.
To find the equation of the tangent line at x=4, we use the point-slope form of a line y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
Using the point (4, f(4)) and the slope -17, we get y - f(4) = -17(x - 4).
Simplifying, we have y - (-2(4)² - 4√(4)) = -17(x -4).
This can be further simplified to y + 32 = -17(x - 4).
Therefore, the equation of the tangent line at x=4 is y = -17x + 96.