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For all real numbers x, let the function...

For all real numbers x, let the function g be defined by `g(x)=p(x-h)^(2)+k`, wher p, h and k are constants with `p, k gt0`. Which of the following CANNOT be true?

A

`g(7)=-h`

B

`g(7)=2`

C

`g(0)=-2`

D

`g(0)=2`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze the function \( g(x) = p(x - h)^2 + k \), where \( p, h, k \) are constants with \( p > 0 \) and \( k > 0 \). We want to determine which of the provided options cannot be true. ### Step-by-Step Solution: 1. **Understanding the Function**: The function \( g(x) = p(x - h)^2 + k \) is a quadratic function that opens upwards because \( p > 0 \). The term \( (x - h)^2 \) is always non-negative (i.e., \( (x - h)^2 \geq 0 \)), which means \( g(x) \) will always be at least \( k \). 2. **Finding the Minimum Value**: Since \( (x - h)^2 \) is minimized at \( x = h \), the minimum value of \( g(x) \) occurs at: \[ g(h) = p(0) + k = k \] Given that \( k > 0 \), the minimum value of \( g(x) \) is \( k \), which is positive. Therefore, \( g(x) \) can never be less than \( k \) and is always positive. 3. **Analyzing the Options**: Now we will analyze each option to see if it can be true given that \( g(x) \) is always positive. - **Option 1**: \( g(7) = -h \) - Since \( h \) can be any real number, \( -h \) can be positive or negative. Therefore, this option can potentially be true depending on the value of \( h \). - **Option 2**: \( g(7) = 2 \) - This is a positive value. Since \( g(x) \) can take positive values, this option can be true. - **Option 3**: \( g(0) = -2 \) - This is a negative value. Since we established that \( g(x) \) is always positive, this option cannot be true. - **Option 4**: \( g(0) = 2 \) - This is a positive value. Therefore, this option can be true. 4. **Conclusion**: The only option that cannot be true is **Option 3**: \( g(0) = -2 \). ### Final Answer: The option that cannot be true is: **Option 3: \( g(0) = -2 \)**.
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