The key concept to know for this question is to truly understand what each functions mean and what the derivative of functions mean.
The first derivative of the function ( y' = 3x^2 - 54x + 29 ) represents the equation to find slope. Hence, in order to solve for the smallest possible slope, we need to find the minimum point of this particular graph. As this equation is a positive quadratic expression, we know that the absolute minimum point would occur at relative minimum. So, we take the second derivative.
The second derivative of the function is y'' = 6x - 54. The extrema happens when this equation is equal to zero, which would happen when x = 9. Hence, the slope for a tangent to the graph is the smallest when x = 9.
This particular slope is ( 3*(9)^2 - 54*9 + 29 = -214). Hence, the smallest possible slope for a tangent to the graph of the equation is -214.
Hi,
Thank you for your question.
The key concept to know for this question is to truly understand what each functions mean and what the derivative of functions mean.
The first derivative of the function ( y' = 3x^2 - 54x + 29 ) represents the equation to find slope. Hence, in order to solve for the smallest possible slope, we need to find the minimum point of this particular graph. As this equation is a positive quadratic expression, we know that the absolute minimum point would occur at relative minimum. So, we take the second derivative.
The second derivative of the function is y'' = 6x - 54. The extrema happens when this equation is equal to zero, which would happen when x = 9. Hence, the slope for a tangent to the graph is the smallest when x = 9.
This particular slope is ( 3*(9)^2 - 54*9 + 29 = -214). Hence, the smallest possible slope for a tangent to the graph of the equation is -214.
Does this answer your question?