Sketching a Graph Sketch a graph of a differentiable function f that satisfies the following conditions and has x = 2 as its only critical number. f'(x) < 0 for x < 2 f'(x) > 0 for x > 2 lim f(x) = 6 lim f(x) = 6
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- Analyzing critical points Find the critical points of the following functions.Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, a local minimum,or a saddle point. If the Second Derivative Test is inconclusive,determine the behavior of the function at the critical points. ƒ(x, y) = yex - eyAnalyzing critical points Find the critical points of the following functions.Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, a local minimum,or a saddle point. If the Second Derivative Test is inconclusive,determine the behavior of the function at the critical points. ƒ(x, y) = (4x - 1)2 + (2y + 4)2 + 187. Analyzing a Critical Number A differentiable function $f$ has one critical number at $x=5$. Identify the relative extrema of $f$ at the critical number when $f^{\prime}(4)=-2.5$ and $f^{\prime}(6)=3$
- First Derivative Testa. Locate the critical points of f.b. Use the First Derivative Test to locate the local maximum andminimum values.c. Identify the absolute maximum and minimum values of the functionon the given interval (when they exist).Consider a differentiable function f with domain R and derivativesf'(x)=-aebx(1+bx) and f"(x)=-abebx(2+bx) , with a and b nonzero real numbers.The function has only one critical point x=-1/b and a local maximum at x=-1/bUse the Second Derivative test to find the value(s) of a and bCalculus I In the exercise f(x)= cos x + sin x; [0,2pi], find the following 1.) Search for critical points2.) Search if it grows or decreases3.) Search for local maximum and minimum
- Analyzing critical points Find the critical points of the following functions.Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, a local minimum,or a saddle point. If the Second Derivative Test is inconclusive,determine the behavior of the function at the critical points.Find the absolute maximum and minimum points, if they exist, of f(x)=x3+x2-x+1 the interval [-2, 1/2] show work to find all critical points in the intervalFinding the critical point(s) of f(x) on the given internal. f(x)=(2^x)sin(x) on [-2,6]
- Using the First Derivative Test proved in the videos, prove the following version of the First Derivative Test: If f′ is continuous on the interval [a,b] and if f has exactly one critical point c then f has a maximum at c if f′(a′)>0 and f′(b′)<0 for some a′ and b′ such that a<a′<c<b′<b.1. Find all critical points of f. 2. Classify each critical point that you found in part (a) as a local maximum, local minimum, or saddle point. 3. Find the absolute maximum and minimum of f on the region bounded by the x- and y-axes along with the curve xy = 18the continuous function has a critical point. (a)Is the critical point a local maximum or a local minimum? (b)Sketch the graph near the critical point. Label the coordinates of the critical point. 1. f(1) = 5, f ′(1) = 0, f ″(1) = −2 2. h(2) = −5, h′(2) = 0, h″(2) = −4