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Let P(x)=x⁴+ax³+bx²+cx+d, and suppose that x=0 is the only real solution of P'(x)=0. If P(-1)<P(1), then on the interval [-1,1] which statement is true?
- P(-1) is not the minimum, but P(1) is the maximum value of P.
- P(-1) is the minimum, but P(1) is not the maximum value of P.
- P(-1) is neither the minimum nor is P(1) the maximum value of P.
- P(-1) is the minimum and P(1) is the maximum value of P.
Correct answer: P(-1) is not the minimum, but P(1) is the maximum value of P.
Solution
Since x=0 is the only root of P'(x)=4x^3+3ax^2+2bx+c, we need c=0 and 4x^2+3ax+2b>0, so P'(x)=x*(positive). Thus P decreases on [-1,0], increases on [0,1] (min at 0, not at -1). Given P(-1)<P(1), the maximum on [-1,1] is P(1). So P(-1) is not the minimum but P(1) is the maximum.
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