The mean value of a function $f(x)$ from $x = a$ to $x = b$ is given by:
Options:
(A) [f(a)+f(b)]/2
(B) [f(a)+2f((a+b)/2)+f(b)]/4
(C) ∫ₐᵇ f(x)dx
(D) [∫ₐᵇ f(x)dx]/(b−a)
Official Answer Key:
D
Step-by-Step Solution & Explanation
By definition, the mean (average) value of a continuous function over [a, b] is the total 'area' under the curve divided by the width of the interval: f_mean = (1/(b−a))·∫ₐᵇ f(x)dx. Option (C) is only the integral (total area, not an average), and options (A)/(B) are numerical-integration approximations (trapezoidal-type and Simpson-type averages of end/mid ordinates) rather than the exact analytical definition of the mean value.
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