Final answer:
To find the point on the parabola y=-x³+3x+4 with a tangent slope of 5, calculate the derivative to get y' = -3x² + 3, set it equal to 5, solve for x, and then use the x-value to find the corresponding y-value on the parabola.
Step-by-step explanation:
To find the point on the parabola y=-x³+3x+4 where the slope of the tangent line is 5, we need to first find the derivative of the equation to determine the slope function. The derivative, y', of the given parabola is y' = -3x² + 3. We set this equal to 5, the given slope, to solve for x: -3x² + 3 = 5.
Upon simplifying, we get a quadratic equation: -3x² - 2 = 0. Solving this gives us the x-coordinate where the slope is 5. Finally, we plug this x-coordinate back into the original equation to find the corresponding y-coordinate, giving us the point on the parabola where the slope of the tangent line is 5.