Maths · Applications of derivatives: Rate of change of quantities, monotonic-Increasing and decreasing functions, Maxima and minima of functions of one variable
If =(x-a)^{2 n}(x-b)^{2 m+1} \) where then
If \( f(x)=(x-a)^{2 n}(x-b)^{2 m+1} \) where \( m \cdot n \in N, \) then
- A. \( x=a \) is a point of minimum
- B. \( x=a \) is a point of maximum
- C. \( x=a \) is not a point of maximum or minimum
- D. No value of k satisfies the requirement
Step-by-step solution
Since the exponent of (x-a) is even (2n), the function f(x) does not change sign across x=a. The other factor (x-b)^{2m+1} is continuous and non-zero at x=a (assuming a≠b). Thus, f(x) has a local extremum at x=a. The leading coefficient of the polynomial is positive, and as x→±∞, f(x)→±∞ respectively. For a root with even multiplicity, the graph touches the x-axis and turns, typically resulting in a local minimum. Therefore, x=a is a point of minimum.
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