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Question

Let x be a positive real number. Find the minimum value of 4x^5 + 5x^-4.

Accepted Answer

By AM-GM, align* 4x^5 + 5x^-4 &= x^5 + x^5 + x^5 + x^5 + x^-4 + x^-4 + x^-4 + x^-4 + x^-4 & 9 [9](x^5)^4 (x^-4)^5 &= 9. align*Equality occurs when x = 1, so the minimum value is 9.