Slopes and Tangent Lines

b. Smallest slope Wh...

Question

Slopes and Tangent Lines

b. Smallest slope What is the smallest slope on the curve? At what point on the curve does the curve have this slope?

Accepted Answer

Welcome back, everyone. Consider the curve defined by Y equals x cubed minus 8 x + 5. Determine the smallest slope of the curve and the point at which it occurs. So we are given our curve, and we want to identify the slope of the tangent line so we can do that by differentiating the function. So let's differentiate X cubed minus 8 x + 5. If we apply the power rule, we're going to get 3 x quad minus 8. So that's the equation of a slope at any X, right? And we want to identify the smallest slope. So essentially we have the equation of a slope in a general form, and then we want to minimize it. So if we want to perform an optimization to this expression, we have to identify as derivative. Simply speaking, we're looking for the second derivative, which is equal to the derivative of 3 X2 minus 8. And applying the power rule, we get 6 X. So now, we want to prove that there is a local minimum within this function, and specifically we're looking for the absolute minimum, right? But we have an open interval for all real numbers. So, Essentially what we have to do is identify the critical points. We're going to set the 2nd derivative equal to 0, which means that 6 X is equal to 0 and therefore our critical point is x equals 0. And now let's prove that this is a local minimum. So what we're going to do is simply plot a number line, add 0, and divide the our interval for all real numbers into two parts below 0 and above 0. We want to evaluate the second derivative at any point that is lower than 0, let's say -1, which gives us 6 multiplied by -1 and that's -6. So if we get a negative value, this means that the slope. Is decreasing. And then we want to evaluate the second derivative at any point above 0, such as 1. Which gives us 6 multiplied by 1 and that's 6, that's positive, right? So we are changing sign from negative to positive meaning past 0 hour. Function of the slope starts to increase, so indeed we have a local minimum at x equals 0, right? Hold on, so we have our proof, and now what we're going to do is simply evaluate the coordinates of the point. We know that X equals 0 and Y. And X equals 0 is going to be equal to 0 cubed minus 8 multiplied by 0 + 5 based on the original expression of the curve, right? So 0 minus 0 + 5 gives us 5. Therefore, the coordinates are 0.5, and now we can determine that slope, right? What is the smallest slope? We have to recall that the equation of the slope is Y equals 3x2 minus 8. So we simply want to evaluate Y at 0 because this is where the smallest slope occurs, and we get 3 multiplied by 02 minus 8, which is -8. So, The smallest slope is -8 and it occurs at a 0.0.5. That's our final answer and thank you for watching.

Slopes and Tangent Lines

b. Smallest slope What is the smallest slope on the curve? At what point on the curve does the curve have this slope?

Key Concepts

Derivative

Critical Points

Second Derivative Test

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