Math 2A Single Variable Calculus Homework Questions 44Applications of Derivatives4.1Maximum and Minimum Values1. Find the critical points off(x) =x3+6x2-15x.2. Find the critical points ofh(p) =p-1p2+4.3. Find the critical points ofg(θ) =4θ-tanθ.4. Find the absolute maximum and minimum values off(x) =5+54x-2x3on the interval[0, 4].5. Find the absolute maximum and minimum values off(x) =x3-6x2+5 on the interval[-3, 5].6. Find the absolute maximum and minimum values off(x) =xx2-x+1on the interval[0, 3].7. Find the absolute maximum and minimum values off(t) =3√t(8-t)on the interval[0, 8].8. An object with weightWis dragged along a horizontal plane by a force acting along a ropeattached to the object. If the rope makes an angleθwith the plane, then the magnitude of theforce isF=μWμsinθ+cosθ,whereμis a positive constant called thecoefficient of frictionand 0≤θ≤π/2. Show thatFisminimized when tanθ=μ.9. Show that 5 is a critical number of the functiong(x) =2+ (x-5)3,but thatgdoes not have a local extreme value atx=5.4.2The Mean Value Theorem1. Letf(x) =tanx. Show thatf(0) =f(π)but that there is no numberxin(0,π)such thatf0(c) =0. Why does this not contradict Rolle’s Theorem?2. Verify that the functionf(x) =x3-3x+2 satisfies the hypotheses of the Mean Value Theoremon the interval[-2, 2], and find all numberscthat satisfy the conclusion of the Mean ValueTheorem.3. Verify that the functionf(x) =1/xsatisfies the hypotheses of the Mean Value Theorem on theinterval[1, 3], and find all numberscthat satisfy the conclusion of the Mean Value Theorem.4. Letf(x) =2- |2x-1|. Show that there is no value ofcsuch thatf(3)-f(0) =f0(c)(3-0).Why does this not contradict the Mean Value Theorem?1