Find all the local maxima, local minima, and saddle points of f(x, y) = x³ + 3y² + 3xy – 3y
To find the local maxima, local minima, and saddle points for the function f(x, y) = x³ + 3y² + 3xy – 3y, we first need to compute the critical points. We begin by calculating the first order partial derivatives: fx(x, y) = 3x² + 3y fy(x, y) = 6y + 3x – 3 Next, […]
How to write nine crore five lakh forty one in number?
To write ‘nine crore five lakh forty one’ in numerical form, we break down the components of the phrase: ‘Nine crore’ equals 9,00,00,000 (which is 9 followed by 7 zeros). ‘Five lakh’ equals 5,00,000 (which is 5 followed by 5 zeros). ‘Forty one’ is simply 41. Now, we need to add these values together: 9,00,00,000 […]
How to Find the HCF of 65 and 117 and Determine Integral Values for m and n?
To find the highest common factor (HCF) of 65 and 117, we can use the method of prime factorization or the Euclidean algorithm. Here, we’ll use the Euclidean algorithm for simplicity. Step 1: Apply the Euclidean algorithm. We divide the larger number (117) by the smaller number (65) and find the remainder: 117 ÷ 65 […]
Find the midpoint of the line segment whose endpoints are (3, 10) and (7, 8)
To find the midpoint of a line segment with given endpoints, you can use the midpoint formula. The midpoint (M) can be calculated using the following formula: M = ((x1 + x2) / 2, (y1 + y2) / 2) Here, (x1, y1) = (3, 10) and (x2, y2) = (7, 8). Now, plug in the […]
If the volume of a sample of water is 25 millilitres (ml), what is the volume of this sample in litres (l)?
To convert millilitres to litres, we use the conversion factor that 1 litre is equal to 1000 millilitres. Therefore, to convert 25 millilitres to litres, we divide the number of millilitres by 1000. Here’s the calculation: 25 ml ÷ 1000 = 0.025 l So, the volume of the sample is 0.025 litres. This conversion is […]
What is the formula for the area of a rectangle?
The formula for the area of a rectangle is A = l × w, where A represents the area, l is the length, and w is the width of the rectangle. To understand this formula, consider that the area is essentially measuring the amount of space contained within the rectangle. By multiplying the length by […]
If n = 15 and p = 0.4, what is the standard deviation of the binomial distribution?
To find the standard deviation of a binomial distribution, we can use the formula: Standard Deviation (σ) = √(n * p * (1 – p)) In this case, we have: n = 15 (the number of trials) p = 0.4 (the probability of success) First, we calculate (1 – p): (1 – p) = 1 […]
The sum of two numbers is 14 and their difference is 6 what are the two numbers?
To find the two numbers, we can set up a system of equations based on the information given. Let’s denote the two numbers as x and y. The first piece of information tells us that the sum of the two numbers is 14, so we can write the equation: x + y = 14 The […]
Which of the following represents the zeros of f(x) = x³ + 3x² + 2x + 6?
To find the zeros of the polynomial function f(x) = x³ + 3x² + 2x + 6, we need to solve the equation f(x) = 0. This means we are looking for values of x that make the polynomial equal to zero. The polynomial can be factored or solved via various methods such as synthetic […]
How to Write the Sum or Difference in the Standard Form a + bi: 3, 3i, 6, i?
To find the sum or difference of the complex numbers 3, 3i, 6, and i, we first need to identify their real and imaginary parts. The numbers can be broken down as follows: 3 is a real number (3 + 0i) 3i is an imaginary number (0 + 3i) 6 is also a real number […]