If f(x) = 2x2 - 3x + 1, find the value of 2f(0) + f(-1) - 3f(f(0)). 2. If 10x + 10y = 2, then the smallest possible value of (x+y) is

Here are the solutions to the problems: 1. If f(x) = 2x^2 - 3x + 1, find the value of 2f(0) + f(-1) - 3f(f(0)). Step 1: Calculate f(0). f(0) = 2(0)^2 - 3(0) + 1 = 0 - 0 + 1 = 1 Step 2: Calculate f(-1). f(-1) = 2(-1)^2 - 3(-1) + 1 = 2(1) + 3 + 1 = 2 + 3 + 1 = 6 Step 3: Calculate f(f(0)). Since f(0)=1, we need to find f(1). f(1) = 2(1)^2 - 3(1) + 1 = 2 - 3 + 1 = 0 Step 4: Substitute the calculated values into the expression. 2f(0) + f(-1) - 3f(f(0)) = 2(1) + 6 - 3(0) = 2 + 6 - 0 = 8 The value is 8. The correct option is (B). B 2. If _10x + _10y = 2, then the smallest possible value of (x+y) is Step 1: Use the logarithm property _a M + _a N = _a (MN). _10(xy) = 2 Step 2: Convert the logarithmic equation to an exponential equation. xy = 10^2 xy = 100 Step 3: To find the smallest possible value of x+y for positive x and y (since they are arguments of a logarithm), we use the AM-GM inequality, which states that for non-negative numbers a and b, (a+b)/(2) sqrt(ab). (x+y)/(2) sqrt(xy) Step 4: Substitute xy=100 into the inequality. (x+y)/(2) sqrt(100) (x+y)/(2) 10 x+y 20 The equality holds when x=y. If x=y, then x · x = 100 x^2 = 100. Since x>0, x=10. Thus, y=10. The smallest value of x+y is 10+10=20. The correct option is (C). C 3. If y = sqrt(x) e^x, find (dy)/(dx). Step 1: Identify the function as a product of two functions: u(x) = sqrt(x) and v(x) = e^x. We will use the product rule for differentiation: (d)/(dx)(uv) = u(dv)/(dx) + v(du)/(dx). Step 2: Find the derivative of u(x) = sqrt(x) = x^1/2. (du)/(dx) = (1)/(2)x^(1)/(2)-1 = (1)/(2)x^-(1)/(2) = (1)/(2sqrt(x)) Step 3: Find the derivative of v(x) = e^x. (dv)/(dx) = e^x Step 4: Apply the product rule. (dy)/(dx) = (sqrt(x))(e^x) + (e^x)((1)/(2sqrt(x))) Step 5: Factor out e^x and simplify. (dy)/(dx) = e^x (sqrt(x) + (1)/(2sqrt(x))) This matches option (B). The correct option is (B). B Send me the next one 📸