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To find the coordinates of the turning point for the function f(x) = x³ + 3x + 1, we first need to calculate the derivative of the function. The turning points occur where the derivative is equal to zero, as these are the locations where the function may change direction (i.e., from increasing to decreasing … Read more

In a geometric sequence, each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In your case, the first term a1 is 3, and the common ratio r is -1. The general formula for the n-th term of a geometric sequence can be expressed … Read more

To multiply expressions with the same base, you can simply add the exponents. In this case, we need to multiply x^{1/3} by x^{3/7}. The rule for multiplying exponents is as follows: If you have a^m multiplied by a^n, the result is a^{m+n}. So, let’s add the exponents: Start by rewriting the expression: x^{1/3} imes x^{3/7} … Read more

To determine the values of m y 26, m y 34, m y 60, and m y 120 based on the given conditions of m x 34 and m z 26, we need to understand the relationship between these variables. Firstly, it’s important to note that ‘m’ generally refers to a variable and ‘x’, ‘y’, … Read more

To combine the expression 3 ln(x) + 2 ln(c) into a single natural logarithm, we can utilize the properties of logarithms. Specifically, we will be using the power rule and the product rule of logarithms. Power Rule: The power rule states that n ln(a) = ln(a^n). This means you can move the coefficient of the … Read more

To divide the polynomial expression 4×3 + 2×2 + 3x + 4 by x4, we need to perform polynomial long division or simplify the division. Since we are dividing by a monomial, we can simplify the expression by dividing each term of the polynomial by x4. Here’s how it works step by step: Write the … Read more

The equation of a parabola can be derived based on its geometric properties. In this case, we have a parabola with a vertex located at the origin (0, 0) and a focus located at the point (0, 2). Firstly, since the focus is positioned above the vertex along the y-axis, this indicates that the parabola … Read more

To find the range of the function y = 12sin(x) + p, we need to understand how the sine function behaves and how it is affected by the transformations applied to it. The sine function, sin(x), oscillates between -1 and 1. Therefore, when we multiply it by 12, the range of 12sin(x) becomes: -12 <= … Read more

To find the length of a chord that connects two perpendicular radii of a sphere, we can use some basic geometry. 1. **Understanding the Setup**: The two radii, being perpendicular, form a right triangle with the chord as the hypotenuse. Each radius measures 6 inches, and they meet at the center of the sphere. 2. … Read more

To find the values of sin 2theta and cos 2theta when cos theta is given as 12/13, we can use the double angle formulas in trigonometry: sin 2theta = 2 * sin theta * cos theta cos 2theta = cos2theta – sin2theta First, we need to determine the missing value: sin theta. Using the Pythagorean … Read more