Final answer:
To determine when a function is increasing, we find the intervals where the derivative is positive. For each function provided, I will show you how to find those intervals.
Step-by-step explanation:
To determine when a function is increasing, we need to find the interval where the derivative of the function is positive. For f(x) = x² + 3x - 5, we can find the derivative by taking the derivative of each term separately. The derivative of x² is 2x, the derivative of 3x is 3, and the derivative of -5 is 0. Therefore, the derivative of f(x) is 2x + 3. To find when the derivative is positive, we set it greater than zero and solve: 2x + 3 > 0. Solving for x, we get x > -3/2. So, f(x) is increasing on the interval (-3/2, ∞).
For f(x) = -2x³ + 6x² - 4x + 1, we can find the derivative by taking the derivative of each term separately. The derivative of -2x³ is -6x², the derivative of 6x² is 12x, the derivative of -4x is -4, and the derivative of 1 is 0. Therefore, the derivative of f(x) is -6x² + 12x - 4. To find when the derivative is positive, we set it greater than zero and solve: -6x² + 12x - 4 > 0. We can use the quadratic formula to solve for x, which gives us two solutions: x ≈ 0.33 and x ≈ 1.00. So, f(x) is increasing on the intervals (-∞, 0.33) and (1.00, ∞).
For f(x) = eˣ - x², we can find the derivative by using the rules of differentiation. The derivative of eˣ is eˣ, and the derivative of -x² is -2x. Therefore, the derivative of f(x) is eˣ - 2x. To find when the derivative is positive, we set it greater than zero and solve: eˣ - 2x > 0. This equation does not have an algebraic solution, so we can use numerical methods or a calculator to find the intervals where the derivative is positive. From the graph or calculator, we can see that the f(x) is increasing on the interval (-∞, 2.08).
For f(x) = sin(x) + cos(x), we can find the derivative by using the rules of differentiation. The derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x). Therefore, the derivative of f(x) is cos(x) - sin(x). To find when the derivative is positive, we set it greater than zero and solve: cos(x) - sin(x) > 0. This equation does not have an algebraic solution, so we can use a calculator or graphing software to find the intervals where the derivative is positive. From the graph or calculator, we can see that f(x) is increasing on the intervals (-∞, 0.785) and (5.498, ∞).