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Math  /  Calculus

Question11st&2nd 1 \quad 1^{s t} \& 2^{\text {nd }} Derivative test
For the following functions defined below, (a) Locate the critical points of ff. (b) Use the First Derivative Test to locate the local maximum and minimum values. (c) Identify the inflection points. (d) Use the Second Derivative Test to locate the local maximum and minimum values. (i) f(x)=2x3+5x2−10xf(x)=2 x^{3}+5 x^{2}-10 x (ii) f(x)=x2−3x+2f(x)=x^{2}-3 x+2 (iii) f(x)=x2−x3f(x)=x^{2}-\sqrt[3]{x}

Studdy Solution
(i) For f(x)=2x3+5x2−10xf(x) = 2x^3 + 5x^2 - 10x: Critical points at x=−5±856x = \frac{-5 \pm \sqrt{85}}{6}, inflection point at x=−56x = -\frac{5}{6}. (ii) For f(x)=x2−3x+2f(x) = x^2 - 3x + 2: Critical point (and local minimum) at x=32x = \frac{3}{2}, no inflection points. (iii) For f(x)=x2−x3f(x) = x^2 - \sqrt[3]{x}: Critical point (and local minimum) at x=(16)35x = \left(\frac{1}{6}\right)^{\frac{3}{5}}, no inflection points.

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