The tangents to the curve with equation y = x^3-3x at the points A and B with x coordinates −1 and 4 respectively meet at the point C. Find the coordinates of the point C.

Step-by-step explanation:

We need to find the equation of the tangent lines of Points A and B.

Differentiate the equation:

Point A has an x-coordinate of -1. Hence, the slope of its tangent line is:

Find the y-coordinate of Point A using the original equation:

Hence, Point A is (-1, 2).

Thus, the tangent line at Point A is:

Simplify:

Point B has an x-coordinate of 4. Hence, the slope of its tangent line is:

Find the y-coordinate of Point B:

Thus, Point B is at (4, 52).

So, the tangent line at Point B is:

Simplify:

Point C occurs at the intersections of the tangent lines of Points A and B. Set the two equations equal to each other and solve for x:

We acquire:

Since one of our equations is y = 2, the y-coordinate is 2.

Hence, Point C is:

Step-by-step explanation:

y=x³-3x

so coordinates of C are (3,2)

d. 24+-0.114

step-by-step explanation:

the question requires us to calculate the approximate 95% confidence interval for the mean number of ounces of ketchup per bottle in the sample given the following statistics;

sample mean = 24

standard deviation, σ = 0.4

sample size , n = 49

the confidence interval for a population mean is calculated using the formula;

sample mean ± z-score*(σ/sqrt(n))

the z-score associated with a 95% confidence interval, from the standard normal tables, is;

1.96

substituting these values into the formula;

24 ± 1.96*(0.4/7)

= 24 ± 0.112

from the alternatives given, choice d is closest to the above expression;