This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Let's analyze function (c) h(x) = x^3 - 2x^2 - 15x. Step 1: Find the intercepts with the axes. To find the y-intercept, set x=0: h(0) = (0)^3 - 2(0)^2 - 15(0) = 0 The y-intercept is (0,0). To find the x-intercepts, set h(x)=0: x^3 - 2x^2 - 15x = 0 Factor out x: x(x^2 - 2x - 15) = 0 Factor the quadratic expression: x(x-5)(x+3) = 0 This gives the x-intercepts: x=0, x=5, and x=-3. The x-intercepts are (0,0), (5,0), and (-3,0). Step 2: Find the stationary points. First, find the first derivative h'(x): h'(x) = (d)/(dx)(x^3 - 2x^2 - 15x) = 3x^2 - 4x - 15 Set h'(x)=0 to find the x-coordinates of the stationary points: 3x^2 - 4x - 15 = 0 Using the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a: x = -(-4) ± sqrt((-4)^2 - 4(3)(-15))2(3) x = 4 ± sqrt(16 + 180)6 x = 4 ± sqrt(196)6 x = (4 ± 14)/(6) The x-coordinates are: x_1 = (4 + 14)/(6) = (18)/(6) = 3 x_2 = (4 - 14)/(6) = (-10)/(6) = -(5)/(3) Now, find the corresponding y-coordinates by substituting these values back into h(x): For x=3: h(3) = (3)^3 - 2(3)^2 - 15(3) = 27 - 2(9) - 45 = 27 - 18 - 45 = 9 - 45 = -36 Stationary point 1: (3, -36). For x=-(5)/(3): h(-(5)/(3)) = (-(5)/(3))^3 - 2(-(5)/(3))^2 - 15(-(5)/(3)) h(-(5)/(3)) = -(125)/(27) - 2((25)/(9)) + 25 h(-(5)/(3)) = -(125)/(27) - (50)/(9) + 25 To combine these, use a common denominator of 27: h(-(5)/(3)) = -(125)/(27) - (50 × 3)/(9 × 3) + (25 × 27)/(27) h(-(5)/(3)) = (-125 - 150 + 675)/(27) = 400