Question

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question 4 (4 points)
what are the local maxima for $f(x) = x^4 - 8x^2 + 10$?
○ (0, 10)
○ (2, 10) and (-2, 10)
○ no local maxima
○ (2, -6) and (-2, -6)

Explanation:

Step1: Find first derivative

$f'(x) = 4x^3 - 16x$

Step2: Find critical points

Set $f'(x)=0$:
$4x^3 - 16x = 0$
$4x(x^2 - 4) = 0$
$4x(x-2)(x+2) = 0$
Critical points: $x=0, x=2, x=-2$

Step3: Find second derivative

$f''(x) = 12x^2 - 16$

Step4: Test critical points

For $x=0$: $f''(0) = 12(0)^2 -16 = -16 < 0$ (local max)
For $x=2$: $f''(2) = 12(4) -16 = 32 > 0$ (local min)
For $x=-2$: $f''(-2) = 12(4) -16 = 32 > 0$ (local min)

Step5: Find local max value

$f(0) = 0^4 -8(0)^2 +10 = 10$

Answer:

(0, 10)