Let f(x)=2x−x2,x∈R. If m and n are respectively the number of points at which the curves y=f(x) and y=f′(x) intersects the x-axis, then the value of m+n is
Question
Let f(x)=2x−x², x∈R. If m and n are respectively the number of points at which the curves y=f(x) and y=f′(x) intersects the x-axis, then the value of m+n is
Solution
To solve this problem, we need to find the roots of the functions f(x) and f'(x), which represent the points where the curves intersect the x-axis.
Step 1: Find the roots of f(x) f(x) = 2x - x^2 Setting f(x) to 0, we get: 0 = 2x - x^2 This can be factored to: 0 = x(2 - x) Setting each factor equal to zero gives the roots x = 0 and x = 2. So, m = 2.
Step 2: Find the derivative of f(x) and its roots f'(x) = 2 - 2x Setting f'(x) to 0, we get: 0 = 2 - 2x Solving for x gives x = 1. So, n = 1.
Step 3: Find m + n m + n = 2 + 1 = 3.
So, the value of m + n is 3.
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