Question

exercises 4.3 maxima and minima
score: 4/30 answered: 4/14
question 5
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consider the function ( f(x)=-5 x^{2}+2 x - 2 ). ( f(x) ) has a critical point at ( x = a ).
find the value of ( a ):
( a=)
at ( x = a ), does ( f(x) ) have a local min, a local max, or neither? type in your answer as lmin, lmax, or neither.
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Explanation:

Step1: Find the derivative of the function

The derivative of \(f(x)=-5x^{2}+2x - 2\) using the power rule \((x^n)^\prime=nx^{n - 1}\) is \(f^\prime(x)=-10x + 2\).

Step2: Set the derivative equal to zero to find critical points

Set \(f^\prime(x)=0\), so \(-10x+2 = 0\).
Solve for \(x\):

$$
LATEXBLOCK0
$$

Step3: Use the second - derivative test

The second derivative \(f^{\prime\prime}(x)=-10\).
Since \(f^{\prime\prime}(A)=f^{\prime\prime}(\frac{1}{5})=-10<0\), by the second - derivative test (if \(f^{\prime}(c) = 0\) and \(f^{\prime\prime}(c)<0\), then \(x = c\) is a local maximum).

Answer:

\(A = 0.2\)
\(LMAX\)