+89 (d) Differentiate each of the following functions and find the value of the derivative when x = 1:
(i) f(x) = 1−x3(5 ln(x) + 2),
(ii) g(x) = 1 + xex 1 + x2 ,
(iii) h(x) = √1 + x2.
(e) Let f(x) = x3 + 3x2 −24x + 5.
(i) Find f0(x) and f00(x).
(ii) Determine the critical points of f.
(iii) Show that x = −4 is a local maximum of f.
(iv) Show that x = 2 is a local minimum of f
+9664 \(\frac{d}{dx}(1-x^3(5\ln x + 2))\)
\(=\frac{d}{dx}1 - \frac{d}{dx} 5x^3\ln x +\frac{d}{dx} 2x^3\)
\(=\frac{d}{dx}2x^3-(5x^3)\frac{d}{dx}\ln x - (\ln x)\frac{d}{dx} 5x^3 \)
\(= 6x^2 - (5x^3)(\frac{1}{x})-(\ln x )(15x^2)\)
\(= 6x^2-5x^2-15x^2\ln x\)
\(= x^2-15x^2\ln x\)
.+9664 \(\frac{d}{dx} (1 + xe^x (1+x^2))\)
\(= \frac{d}{dx} 1 + (1+x^2)\frac{d}{dx}xe^x+(xe^x)\frac{d}{dx}(1+x^2)\)
\(= (1+x^2)(e^x)(1+x)+(x)(e^x)(2x)\)
\(=(e^x)((1+x^2)(1+x)+2x^2)\)
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