Math  /  Calculus

Question(b) Given y=32x5y=3^{2 x-5}. Find d2ydx2\frac{d^{2} y}{d x^{2}} when x=3x=3. Give the answer in the form of logarithm.
3. The function f(x)=x36x2+9x3f(x)=x^{3}-6 x^{2}+9 x-3 is defined on the interval [0,5][0,5]. Find the [4 marks] critical points of f(x)f(x) on this interval and determine whether the critical points are local minimum or maximum. [7 marks]

Studdy Solution
Determine the nature of the critical points using the second derivative test.
Find the second derivative:
f(x)=6x12 f''(x) = 6x - 12
Evaluate at x=1 x = 1 :
f(1)=6(1)12=6 f''(1) = 6(1) - 12 = -6
Since f(1)<0 f''(1) < 0 , x=1 x = 1 is a local maximum.
Evaluate at x=3 x = 3 :
f(3)=6(3)12=6 f''(3) = 6(3) - 12 = 6
Since f(3)>0 f''(3) > 0 , x=3 x = 3 is a local minimum.
The second derivative of y y when x=3 x = 3 is 12ln2(3) 12 \ln^2(3) . The critical points of f(x) f(x) on [0,5][0,5] are x=1 x = 1 (local maximum) and x=3 x = 3 (local minimum).

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