Question

exercises 4.3 maxima and minima
score: 14/30 answered: 8/14
question 9
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consider the function ( f(x)=1 - 7x^{2}, - 5leq xleq1 ).
the absolute maximum value is
and this occurs at ( x = )
the absolute minimum value is
and this occurs at ( x = )
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Explanation:

Step1: Find the derivative of the function

The derivative of \(f(x)=1 - 7x^{2}\) is \(f^\prime(x)=-14x\).
Set \(f^\prime(x) = 0\), so \(-14x=0\), which gives \(x = 0\).

Step2: Evaluate the function at critical and endpoints

Evaluate \(f(x)\) at \(x=-5\), \(x = 0\), and \(x = 1\).
For \(x=-5\): \(f(-5)=1-7\times(-5)^{2}=1 - 175=-174\).
For \(x = 0\): \(f(0)=1-7\times0^{2}=1\).
For \(x = 1\): \(f(1)=1-7\times1^{2}=1 - 7=-6\).

Answer:

The absolute maximum value is \(1\) and this occurs at \(x = 0\).
The absolute minimum value is \(-174\) and this occurs at \(x=-5\).