Given the graph of y = f'(x), determine: (a) the gradient of the tangent to f(x) at x=0; (b) the value of x where f(x) has a tangent parallel to the tangent at x=0; (c) the intervals where f(x) is dec

AmahleRh🥹❤️, let's knock this out. Step 1: Solve part (a). The gradient of the tangent to the graph of f(x) is given by its derivative, f'(x). We need to find the value of f'(x) at x=0 from the given graph of y=f'(x). From the graph, when x=0, the value of y=f'(x) is 2. f'(0) = 2 The gradient of the tangent to the graph of f(x) at x=0 is 2. Step 2: Solve part (b). A tangent to the curve f(x) will be parallel to the tangent in (a) if it has the same gradient, which is 2. We need to find another value of x where f'(x)=2. Looking at the graph, the parabola y=f'(x) is symmetric. The x-intercepts are at x=-1 and x=5, so the axis of symmetry is at x = (-1+5)/(2) = 2. Since f'(0)=2, by symmetry, f'(x) will also be 2 at x = 2 + (2-0) = 4. So, f'(4)=2. The value of x for which there will be a tangent to the curve f(x) parallel to the tangent in (a) is 4. Step 3: Solve part (c). The function f(x) is decreasing when its derivative f'(x) is negative (f'(x) < 0). From the graph, f'(x) is below the x-axis for x < -1 and for x > 5. So, f(x) is decreasing for x < -1 or x > 5. Step 4: Solve part (d). Turning points of f(x) occur where f'(x) = 0. From the graph, f'(x)=0 at x=-1 and x=5. • At x=-1: f'(x) changes from negative to positive. This indicates a local minimum. • At x=5: f'(x) changes from positive to negative. This indicates a local maximum. The x-coordinates of the turning points of f(x) are x=-1 (local minimum) and x=5 (local maximum). Step 5: Solve part (e). To draw the graph of f(x), we use the following information: • x-intercepts of f(x): -2, 2, 7. • Local minimum at x=-1. • Local maximum at x=5. • f(x) is decreasing for x < -1. • f(x) is increasing for -1 < x < 5. • f(x) is decreasing for x > 5. Based on this, the graph of f(x) would be a cubic function with the following characteristics: 1. It starts from positive infinity, decreasing until it crosses the x-axis at x=-2. 2. It continues decreasing to reach a local minimum at x=-1. 3. It then increases, crossing the x-axis at x=2. 4. It continues increasing to reach a local maximum at x=5. 5. Finally, it decreases, crossing the x-axis at x=7 and continuing towards negative infinity. The sketch would show a curve that starts high, dips to a minimum, rises to a maximum, and then falls. The x-intercepts and turning points are consistent with this shape. Send me the next one 📸